## Hypothesis Testing using Alpha

NFL2000

Using these data, perform an appropriate test of hypothesis, at the alpha = .05 level of significance, to determine whether compensation for players in the NFL is dependent on the team for which they play. What is your conclusion?

NFL salaries and bonuses 2000 season

Salary 49ers Bears Bengals Bills Broncos Browns Buccaneers Cardinals Chargers Chiefs Colts Cowboys Dolphins Eagles Falcons Giants Jaquars Jets Lions Packers Panthers Patriots Raiders Rams Ravens Redskins Saints Seahawks Steelers Titans Vikings Grand Total

0-299999 25 23 21 22 18 31 14 19 18 18 27 26 15 19 19 24 19 16 11 24 22 25 14 16 13 17 16 27 19 21 15 614

300000-599999 34 21 16 30 26 24 25 26 34 33 20 26 37 20 27 32 36 26 34 22 34 27 34 25 36 36 33 21 26 27 33 881

600000-900000 5 8 1 6 2 9 4 2 3 4 1 2 5 7 4 5 5 5 2 2 2 2 4 3 1 4 3 6 2 109

>900000 1 12 11 1 8 5 7 11 4 4 6 4 9 11 8 3 6 6 7 3 3 5 10 5 4 10 5 10 6 4 189

Grand Total 60 61 56 54 58 62 55 60 58 58 57 57 63 55 61 59 59 53 56 58 61 57 55 53 58 60 60 57 58 60 54 1793

## Excel Cake Mix

Copy the Assignment Data (i.e., cake mix data) to Sheet 1, cells A1:P32, of a new Excel workbook.

Select the most appropriate hypothesis test described in Sections 4.3 and 4.5 to evaluate the following null hypothesis using this worksheet: There is no difference in mean grams of sugar between name brands and generic brands of packaged cake mixes. (Note: variable Type, 1=brand name, 2=generic; assume normality and equal variances.) The tests in 4.3 and 4.5 are F Test of Equality of Variance, Independent T Test, dependent t test, wilcoxon matched pair signed ranks test, related samples sign test, and mcnemar test.

Cell A34: Name of most appropriate hypothesis test to evaluate the null hypothesis.

Cells B35:E35: Enter the following labels: Sample Size, Mean, Standard Deviation, Variance.

Cells A36:A38: Enter the following labels: Brand-name, Generic, Sample. (Hint: these are the descriptive statistics for each of the two groups that are compared as well as the overall sample descriptive statistics.)

Cells B36:E38: Enter appropriate formulas to display the required statistics based on adjacent labels.

Cell B40: Enter the following label: Statistics.

Cells A41:A47: Enter the following labels: Pooled variance, Mean difference, SE difference, df, appropriate test statistic (equal variances), p-level (2-tailed), appropriate effect size.

Cells B41:B46: Enter formulas to display the required statistics based on adjacent labels. Format cells to round to two decimal places.

Cell B48: Enter the statistical decision regarding the null hypothesis (i.e., reject or fail to reject the null hypothesis). (Note: Use the .05 significance level.)

Cell B50: Enter test results using APA style (make sure to include informationally adequate statistics). Use adjacent cells as appropriate. (Note: you will not be able to italicize abbreviations and symbols.)

## BMI, Caffeine Consumption, Crossover

Week 4 Problems—30 points

10 points per problem.

• The mean body mass index (BMI) for boys age 12 is 23.6. An investigator wants to test if the BMI is higher in 12-year-old boys living in New York City. How many boys are needed to ensure that a two-sided test of hypothesis has 80% power to detect a difference in BMI of 2 units? Assume that the standard deviation in BMI is 5.7.

Alpha = ________

Z1-α/2= ________

Z1-β = ________

ES = ________

n= ________

2. An investigator wants to estimate caffeine consumption in high school students. How many students would be required to estimate the proportion of students who consume coffee? Suppose we want the estimate to be within 5% of the true proportion with 95% confidence.

Alpha = ________

Z= ________

p= ________

Effect Size = ________

n= ________

3. A crossover trial is planned to evaluate the impact of an educational intervention program to reduce alcohol consumption in patients determined to be at risk for alcohol problems. The plan is to measure alcohol consumption (the number of drinks on a typical drinking day) before the intervention and then again after participants complete the educational intervention program. How many participants would be required to ensure that a 95% confidence interval for the mean difference in the number of drinks is within 2 drinks of the true mean difference? Assume that the standard deviation of the difference in the mean number of drinks is 6.7 drinks.

Z= ________

s= ________

Effect Size = ________

n= ________

## Stock Analyst Statistics

A stock analyst wants to determine whether there is a difference in the mean rate of return for three types of stock: utility, retail, and banking stocks. The following output is obtained:

Picture

a. http://ezto.mheducation.com/13252703214728584664.tp4?REQUEST=SHOWmedia&media=lind16e.png

Using the .05 level of significance, is there a difference in the mean rate of return among the three types of stock?

Yes, since the test statistic is greater than the critical value What is the critical value?

## Chi-Square Tests and Linear Regression

5. At a school pep rally, a group of sophomore students organized a free raffle for prizes. They claimed that they put the names of all the students in the school in the basket and that they randomly drew 36 names out of this basket. Of the prize winners, 6 were freshmen, 14 were sophomores, 9 were juniors, and 7 were seniors. The results do not seem that random to you. You think it is a little fishy that sophomores organized the raffle and also won the most prizes. Your school is composed of 30% freshmen, 25% sophomores, 25% juniors, and 20% seniors.

Freshmen Sophomores Juniors Seniors Totals

Observed 6 14 9 7

Expected

Chi-Square

a. What are the expected frequencies of winners from each class?

b. Conduct a significance test to determine whether the winners of the prizes were distributed throughout the classes as would be expected based on the percentage of students in each group. Report your Chi Square and p values.

c. What do you conclude?

Assume the significance level is 0.05.

14. A geologist collects hand-specimen sized pieces of limestone from a particular

area. A qualitative assessment of both texture and color is made with the following results. Is there evidence of association between color and texture for these limestones? Explain your answer.

Color

Texture Light Medium Dark

Fine 4 20 8

Medium 5 23 12

Coarse 21 23 4

102. Do men and women select different breakfasts? The breakfasts ordered by randomly selected men and women at a popular breakfast place is shown in Table 11.55. Conduct a test for homogeneity at a 5% level of significance.

French Toast Pancakes Waffles Omelettes

Men 47 35 28 53

Women 65 59 55 60

82.

Size (ounces) Cost ($) Cost per ounce

16 3.99

32 4.99

64 5.99

200 10.99

a. Using “size” as the independent variable and “cost” as the dependent variable, draw a scatter plot.

b. Does it appear from inspection that there is a relationship between the variables? Why or why not?

c. Calculate the least-squares line. Put the equation in the form of: ŷ = a + bx

d. Find the correlation coefficient. Is it significant?

e. If the laundry detergent were sold in a 40-ounce size, find the estimated cost.

f. If the laundry detergent were sold in a 90-ounce size, find the estimated cost.

g. Does it appear that a line is the best way to fit the data? Why or why not?

h. Are there any outliers in the given data?

i. Is the least-squares line valid for predicting what a 300-ounce size of the laundry detergent would you cost? Why or why not?

j. What is the slope of the least-squares (best-fit) line? Interpret the slope.

## Percentiles and Hypothesis Testing with Z-Tests

HOMEWORK 6

Percentiles and Hypothesis Testing with Z-Tests

When submitting this file, be sure the filename includes your full name, course and section. Example: HW6_JohnDoe_354B01

Be sure you have reviewed this module/week’s lesson and presentations along with the practice data analysis before proceeding to the homework exercises. Complete all analyses in SPSS, then copy and paste your output and graphs into your homework document file. Answer any written questions (such as the text-based questions or the APA Participants section) in the appropriate place within the same file.

________________________________________

Part I: Concepts

Questions 1-4

________________________________________

These questions are based on the Nolan and Heinzen reading and end-of-chapter questions.

________________________________________

Part I: Questions 1-7

End-of-chapter problems:

Answer the following questions.

If applicable, remember to show work in your homework document for partial credit.

1) What are the 6 steps of hypothesis testing?

(State the 6 steps)

A) Identify the population, comparision distribution, and assumption

B) State the null and research hypothesis

C) Determine the characteristics of the comparision distribution

D) Determine critical values, or cutoff

E) Calculate the test statistics

F) Make a decision

2) Using the z table in Appendix B, calculate the following percentages for a z score of -0.45

2-a) % above this z score: 53% Work: P(z>-.08) =.5319

2-b) % below this z score: 47% Work: P(z<-.08)= .4681

2-c) At least as extreme as this z score (on either side):

94% Work: P(z>.08)+P(z<.08)= 2x .4681=.9362

3) Rewrite each of the following percentages as probabilities, or p levels:

3-a) 5% = .05

3-b) 95% = .95

3-c) 43% = .43

4) If the critical values, or cutoffs, for a two-tailed z test are -2.05 and +2.05, determine whether you would reject or fail to reject the null hypothesis in each of the following cases:

4a) z = 2.23 Since 2.23 is greater than 2.05, we have to fail the null hypothesis.

4b) z = -0.97 Since -.95 is less than -2.05, we have to reject the null hypothesis

5) Imagine a class of twenty-five 12-year-old girls with an average height of 62 inches. We know that the population mean and standard deviation for this age group of girls is m=59 inches, s = 1.5 inches. (Note that this is a z statistic problem.)

5a) Calculate the z statistic for this sample (not the z score). 62-1.5/59=1.02

5b) How does this sample mean compare to the distribution of sample means? In other words, how does the height of the girls in the sample compare to the height of girls in th general population? The majority of the girls are above the average height.

6) For the following scenarios, identify whether the researcher has expressed a directional or a nondirectional hypothesis:

6a) Social media has changed the levels of closeness in long-distance relationships.

Nondirectional

6b) A professor wonders whether students who eat a healthy breakfast score better on exams in morning courses than those who do not eat a healthy breakfast.

Directional

7) For the following scenario, state the null and research hypotheses in both words and symbolic notation. Symbolic notation must include the symbols “” and “” and a comparison operator (=, ≠, <, >, ≤, ≥), as described in Nolan and Heinzen (2014). Remember to consider whether the hypothesis is nondirectional or directional.

Scenario: A professor wonders whether students who eat a healthy breakfast score better on exams in morning courses than those who do not eat a healthy breakfast.

Null Hypothesis (H0): Symbolic Notation HO: m1=m2

Null Hypothesis:

Written Statement Score of a student who eats healthy equals those who don’t.

Research Hypothesis (H1): Symbolic Notation Answer

Research Hypothesis:

Written Statement Answer

________________________________________

Part I: Questions 8a-8g

Fill in the highlighted blanks with the best word or words.

8-a) Values of a test statistic beyond which you reject the null hypothesis are called _Critical values_.

8-b) The _Rejection or Critical Region_ is the area in the tails in which the null can be rejected.

8-c) The probability used to determine the critical values, or cutoffs, in hypothesis testing is known as a _P_ level, also known as alpha.

8-d) If your data differ from what you would expect if chance were the only thing operating, you would call your finding _not significant_.

8-e) A hypothesis test in which the research hypothesis is directional is a(n) _one tailed_ test.

8-f) A hypothesis test in which the research hypothesis specifies that there will be a difference but does not specify the direction of that difference is a(n) _two tailed test.

8-g) If your z-statistic exceeds the critical cutoff, you can _reject_ the null hypothesis.

Part I: Questions 10a-10c

The police department of a major city has found that the average height of their 1,200 officers is 71 inches (in.) with = 2.6 inches. Use the normal distribution and the formulas and steps in this week’s presentations to answer the following questions:

Note: Showing work is required for this section. Remember that it helps to transfer the raw mean and SD from the description above to the standardized curve shown here (though you don’t need to show this). This helps compare raw and z scores and check your work.

10a) What is the z score for an officer who is 72 inches tall? Based on the z score and the z table, what is the officer’s percentile? (Hint: See slide 7 of this week’s related presentation)

Answer (z score): Work (required):

Answer (percentile): Work/reasoning using z table (required):

10b) What is the height (in inches) that marks the 80th percentile for this group of officers? (Hint: See slides 14-16 of this week’s related presentation)

Answer

Work (required):

10c) What percent of officers are between 68 and 72 inches tall? (Hint: See slide 12 of this week’s related presentation)

Answer Work (required):

________________________________________

Part I: Questions 11a-11c

The verbal part of the Graduate Record Exam (GRE) has a of 500 and = 100. Use the normal distribution and the formulas and steps in this week’s presentations to answer the following questions:

Note: Showing work is required for this section. Remember that it helps to transfer the raw mean and SD from the description above to the standardized curve shown here (though you don’t need to show this). This helps compare raw and z scores and check your work.

11a) What is the z score for a GRE score of 583?

What is the percentile rank of this z score? (Hint: See slide 7 of this week’s related presentation)

Answer (z score): Work (required):

Answer (percentile): Work (required):

11b) What GRE score corresponds to a percentile rank of 25%? (Hint: See slide 17 of this week’s related presentation)

Answer Work (required):

11c) If you wanted to select only students at or above the 82nd percentile, what GRE score would you use as a cutoff score (i.e. what GRE score corresponds to this percentile)? (Hint: See slides 14-16 of this week’s related presentation)

Answer Work (required):

________________________________________

Part II: SPSS Analysis

________________________________________

For this section, you will be using last module/week’s data set containing IQ scores.

Open the file; it should also contain the standardized IQ variable you created last module/week.

________________________________________

Part II:

Question 1a & 1b

Use last week’s HW file that you created using IQ scores, and the SPSS reading and presentation from this week. ________________________________________

Using the z-scored IQ variable, create percentile ranks assuming the scores are normally distributed.

Call the new percentile variable “IQ rank.”

1a) List the first 5 IQ ranks from your file (rows 1-5).

Answer:

Row 1: .86, 123

Row 2: .81, 119

Row 3: .50,104

Row 4: .99,145

Row 5: .59,108

1b) Which raw IQ score seems to best divide the top 50% from the bottom 50% of scores?

(This score can be found by looking carefully over the values in the IQ rank column)

104

________________________________________

Part III: SPSS Data Entry and Analysis

________________________________________

There is no Part III material this module/week.

________________________________________

Part IV: Cumulative

Data provided below for respective questions.

________________________________________

Part IV: (Non-SPSS)

Questions 1-4

________________________________________

For a distribution with M = 40 and s = 5:

1) What is the z-score corresponding to a raw score of 32?

-1.6 Work: 32-40/5=-1.6

2) What is the z-score corresponding to a raw score of 50?

2 Work:50-40/5=2

3) If a person has a z-score of 1.8, what is his/her raw score?

32.2 Work: 25+(1.8)(4)=32.2

4) If a person has a z-score of -.63, what is his/her raw score?

22.48 Work: 25+(-.63)(4)=22.48

________________________________________

Part IV: (Non-SPSS)

Question 5-8________________________________________

For the following types of data, state the graph that would be the best choice to display the data.

Two items have more than one correct answer—for these, either answer is acceptable.

5) A nominal independent variable (IV) and a scale dependent variable (DV)

Bar graph

6) One scale variable with frequencies (when you want to see the general shape of the distribution).

Histogram

7) One scale IV and one scale DV

Scatterplot/dot graph

8) One nominal variable broken down into percentages

Pie charts and bar graphs

Submit Homework 6 by 11:59 p.m. (ET) on Monday of Module/Week 6. Remember to name file appropriately.

Done!

## Hypothesis testing using either t or normal distribution

1. A sample of 49 observations is selected from a normal population. The sample mean is 21, and the population standard deviation is 4. Conduct the following test of hypothesis using the 0.05 significance level.

H0 : μ ≤ 20

H1 : μ > 20

a. Is this a one- or two-tailed test?

“One-tailed”-the alternate hypothesis is greater than direction.

“Two-tailed”-the alternate hypothesis is different from direction.

b. What is the decision rule? (Round your answer to 3 decimal places.)

H0, when z >

c. What is the value of the test statistic? (Round your answer to 2 decimal places.)

Value of the test statistic

d. What is your decision regarding H0?

Reject

Do not reject

There is evidence to conclude that the population mean is greater than 20.

e. What is the p-value? (Round your answer to 4 decimal places.)

p-value

2. At the time she was hired as a server at the Grumney Family Restaurant, Beth Brigden was told, “You can average $89 a day in tips.” Assume the population of daily tips is normally distributed with a standard deviation of $2.81. Over the first 40 days she was employed at the restaurant, the mean daily amount of her tips was $92.66. At the 0.01 significance level, can Ms. Brigden conclude that her daily tips average more than $89?

a. State the null hypothesis and the alternate hypothesis.

H0: μ ≥ 89 ; H1: μ < 89

H0: μ ≤ 89 ; H1: μ > 89

H0: μ = 89 ; H1: μ ≠ 89

H0: μ >89 ; H1: μ = 89

b. State the decision rule.

Reject H1 if z > 2.33

Reject H0 if z < 2.33

Reject H0 if z > 2.33

Reject H1 if z < 2.33

c. Compute the value of the test statistic. (Round your answer to 2 decimal places.)

Value of the test statistic

d. What is your decision regarding H0?

Reject H0

Do not reject H0

e. What is the p-value? (Round your answer to 4 decimal places.)

p-value

3. The Rocky Mountain district sales manager of Rath Publishing Inc., a college textbook publishing company, claims that the sales representatives make an average of 41 sales calls per week on professors. Several reps say that this estimate is too low. To investigate, a random sample of 25 sales representatives reveals that the mean number of calls made last week was 43. The standard deviation of the sample is 2.8 calls. Using the 0.010 significance level, can we conclude that the mean number of calls per salesperson per week is more than 41?

H0 : μ ≤ 41

H1 : μ > 41

1. Compute the value of the test statistic. (Round your answer to 3 decimal places.)

Value of the test statistic

4. A United Nations report shows the mean family income for Mexican migrants to the United States is $26,580 per year. A FLOC (Farm Labor Organizing Committee) evaluation of 26 Mexican family units reveals a mean to be $38,900 with a sample standard deviation of $11,054. Does this information disagree with the United Nations report? Apply the 0.01 significance level.

a. State the null hypothesis and the alternate hypothesis.

5. The following information is available

H0: μ =

H1: μ ≠

b. State the decision rule for .01 significance level. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)

Reject H0 if t is not between

and

c. Compute the value of the test statistic. (Round your answer to 2 decimal places.)

Value of the test statistic

5. The following information is available.

H0 : μ ≥ 220

H1 : μ < 220

A sample of 64 observations is selected from a normal population. The sample mean is 215, and the population standard deviation is 15. Conduct the following test of hypothesis using the .025 significance level.

a. Is this a one- or two-tailed test?

Two-tailed test

One-tailed test

b. What is the decision rule? (Negative amount should be indicated by a minus sign. Round your answer to 2 decimal places.)

H0 when z <

c. What is the value of the test statistic? (Negative amount should be indicated by a minus sign. Round your answer to 3 decimal places.)

Value of the test statistic

d. What is your decision regarding H0?

Reject

Do not reject

e. What is the p-value? (Round your answer to 4 decimal places.)

p-value

6. Given the following hypotheses:

H0 : μ ≤ 10

H1 : μ > 10

A random sample of 10 observations is selected from a normal population. The sample mean was 12 and the sample standard deviation 3. Using the .05 significance level:

a. State the decision rule. (Round your answer to 3 decimal places.)

Reject H0 if t >

b. Compute the value of the test statistic. (Round your answer to 3 decimal places.)

Value of the test statistic

7. Given the following hypotheses:

H0 : μ = 400

H1 : μ ≠ 400

A random sample of 12 observations is selected from a normal population. The sample mean was 407 and the sample standard deviation 6. Using the .01 significance level:

a. State the decision rule. (Negative amount should be indicated by a minus sign. Round your answers to 3 decimal places.)

Reject H0 when the test statistic is the interval (

,

).

b. Compute the value of the test statistic. (Round your answer to 3 decimal places.)

Value of the test statistic

Solution Preview

Please check attachment

1. A sample of 49 observations is selected from a normal population. The sample mean is 21, and the population standard deviation is 4. Conduct the following test of hypothesis using the 0.05 significance level.

H0 : μ ≤ 20

H1 : μ > 20

a. Is this a one- or two-tailed test?

“One-tailed”-the alternate hypothesis is greater than direction.

“Two-tailed”-the alternate hypothesis is different from direction.

b. What is the decision rule? (Round your answer to 3 decimal places.)

H0, when z >1.645

c. What is the value of the test statistic? (Round your answer to 2 decimal places.)

Value of the test statistic=(21-20)/[4/sqrt(49)]=1.75

d. What is your decision regarding H0?

Reject

Do not reject

There is evidence to conclude that the population mean is greater than 20.

e. What is the p-value? (Round your answer to 4 decimal places.)

p-value=P(Z>1.75)=0.0401

2. At the time she was hired as a server at the Grumney Family Restaurant, Beth Brigden was told, “You can average $89 a day in tips.” Assume the population of daily tips is normally distributed with a standard deviation of $2.81. Over the first 40 days she was employed at the restaurant, the mean daily amount of her tips was $92.66. At the 0.01 significance level, can Ms. Brigden conclude that her daily tips average more than $89?

a. State the …

## Conduct a two-tailed hypothesis test given the information below

ABC producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

Process A Process B

Sample Mean 2 3

Standard Deviation 1 0.5

Sample Size 12 14

A. What are the assumptions used for this test?

B. The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but assumed equal. What is the critical t value at the 10% level of significance?

C. The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. If we test the null hypothesis at the 5% level of significance, what is the decision?

## hypothesis testing for two means using two paired samples t test

A company is researching the effectiveness of a new website design to decrease the time to access a website. Five website users were randomly selected and their times (in seconds) to access the website with the old and new designs were recorded. The results follow:

User Old Website Design New Website Design

A 30 15

B 45 20

C 25 10

D 32 25

E 28 20

Let α = 0.10. Is the mean time to access the new website design shorter, or is (time for the old design – time for the new design) greater than zero?

a) What is the null hypothesis?

b) What is the alternate hypothesis

c) What is the dregree of freedom 8

d) What is the critical t value at 1% significance level: .10

e) Assume calculated t to be 2.70, at the .01 significance, what would be your decision

## Perform a hypothesis testing for one mean using t distribution

Conduct a one-tailed hypothesis test given the information below.

A manufacturer wants to increase the shelf life of a line of cake mixes. Past records indicate that the average shelf life of the mix is 116 days. After a revised mix has been developed, a sample of nine boxes of cake mix had a mean of 117.333 and a standard deviation of 2.456.

a) At the 0.05 significance level, what is the critical value?

b) State the null and alternative hypothesis.

c) Draw a diagram.

d) Provide the computation of the test statistic.

e) State your decision in terms of the null hypothesis.

f) That is show the 5 steps of hypothesis testing

## Correlation Descriptive Analysis

If you toss a die twice, what is the sample space if you want to restrict the results to only the ones where adding the results of the two throws together equals seven? For example, if you get 1 on the first toss and 6 on the second toss these two tosses meet the criteria.

Which statistics distributions are based on the normal curve?

Which of the following are functions with x denoting the independent variable and y denoting the dependent variable? Choose all that apply.

A. y = f(x)

B. x = f(y)

C. y = a + bx

Which statement(s) is/are false about the Chi Square distribution?

A. As the degrees of freedom increases, the Chi Square distribution approaches a normal distribution.

B. The chi-square distribution is always right skewed, regardless of the value of the degrees of freedom parameter.

C. The chi-square statistic is always positive.

D. As the degrees of freedom increases, the shape of the chi-square distribution becomes more skewed.

## Counting, distribution, paired sample and bi-variate data

If you toss a die twice, what is the sample space if you want to restrict the results to only the ones where adding the results of the two throws together equals seven? For example, if you get 1 on the first toss and 6 on the second toss these two tosses meet the criteria.

Which statistics distributions are based on the normal curve?

Which of the following are functions with x denoting the independent variable and y denoting the dependent variable? Choose all that apply.

A. y = f(x)

B. x = f(y)

C. y = a + bx

Which statement(s) is/are false about the Chi Square distribution?

A. As the degrees of freedom increases, the Chi Square distribution approaches a normal distribution.

B. The chi-square distribution is always right skewed, regardless of the value of the degrees of freedom parameter.

C. The chi-square statistic is always positive.

D. As the degrees of freedom increases, the shape of the chi-square distribution becomes more skewed.

## Parameter and the population of interest for the hypothesis test

Eleven percent of the products produced by an industrial process over the past several months fail to conform to specifications. The company modifies the process to reduce the rate of non-conformities. In a trial run, the modified process produces 16 nonconforming items out of a total of 300 produced. Do these results demonstrate that the modification is effective? Support your conclusion with a test of significance. Assume all conditions have been met.

a. Define the parameter and the population of interest for the hypothesis test.

b. State the type of test you will perform.

c. Write the null and alternate hypotheses.

d. What is the test statistic and the p-value of the test?

f. Construct and interpret a 95% confidence interval for the proportion of nonconforming items for the modified process.

## Wildlife Biologist Studies – Mean Weight

A wildlife biologist captures a certain species of geese in orer to weigh them. Using science, he determines that the weights are normally distributed with a mean of nine pounds and a standard deviation of 1.3 pounds. What is the probability that a goose captured at random will weigh between 8 pounds and 11 pounds?

## Binomial Coefficient and Factorial Notation

What is a binomial coefficient and factorial notation?