## P-value Method: Comparing the BMI’s of Miss America Winners

Please help with the following problem.

Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Use the p-value method for testing hypotheses. Identify the null and alternative hypotheses, test statistic, p-value, critical value, and state the final conclusion.

Listed below are body mass indexes (BMI) for recent Miss America winners.

BMI: 19.5, 20.3, 19.6, 20.2, 17.8, 17.9, 19.1, 18.8, 17.6, 16.8

Use a 0.01 significance level to test the claim that recent winners are from a population with a mean BMI less than 20.16, which was the BMI for winners from the 1920s and 1930s. Do recent winners appear to be significantly different from those in the 1920s and 1930s?

## Hypothesis Thinking in the Court System

In the North American court system, a defendant is assumed innocent until proven guilty. In an ideal world, we would expect that the truly innocent will always go free, whereas the truly guilty ones will always be convicted. Now, let us tackle the following questions.

In the context of the Type I error and Type II error, can you relate a court trial scenario in terms of these two errors?

What would be your ideal situation if you are the defendant?

What would be your ideal situation if you are the prosecuting attorney?

Lastly, what do you think of the scenario of an ideal world where we expect that no innocent will be found guilty and all guilty will be convicted in the context of Type I error and Type II error?

## Manufacturing Models

Outside Magazine tested 10 different models of day hikers and backpacking boots. The following data show the upper support and price for each model tested. Upper support was measured using a rating from 1 to 5, with a rating of 1 denoting average upper support and a rating of 5 denoting excellent upper support.

Manufacturer and Model Support ; Price

Salomon Super Raid 2 ; 120

Merrell Chameleon Prime 3 ; 125

Teva Challenger 3 ; 130

Vasque Fusion GTX 3 ; 135

Boreal Maigmo 3 ; 150

L.L. Bean GTX Super Guide 5 ; 189

Lowa Kibo 5 ; 190

Asolo AFX 520 GTX 4 ; 195

Raichle Mt. Trail GTX 4 ; 200

Scarpa Delta SL M3 5 ; 220

a. Use these data to develop an estimated regression equation to estimate the process of a day hiker and backpacking boot given the upper support training.

b. At the .05 level of significance, determine whether upper support and price are related.

c. Would you feel comfortable using the estimated regression equation developed in part (a) to estimate the price for a day hiker or backpacking boot given the upper support rating?

d. Estimate the price for a day hiker with an upper support rating of 4.

## Medecins Sans Frontières- Statistical Testing

My younger brother had a run in earlier with Medecins Sans Frontières. He narrowly escaped from an adverse verdict by the court. What he wants is that he be left alone to run his small cafe.

He asked my oldest brother if he can conduct a survey for him about justice in the Canadian court.

An initial survey was performed right after Medecins Sans Frontières accused my brother of wrong doing. Out of 1852 customers, 53 were against the aggressive tactics of Medecins Sans Frontières. After my brother was cleared by the court, a follow-up survey was performed. Out of 4699 customers, 1751 said they did not agree with the aggressive tactics of Medecins Sans Frontières.

At the 1% significance level, does the data suggest that a higher percentage of customers were against Medecins Sans Frontières after the court case?

## Statistics: White Blood Cell Counts

White blood cell counts are helpful for assessing liver disease, radiation, bone marrow failure, and infectious diseases. Listed below are recorded white blood cell counts.

A. Use a 0.01 significance level to test the claim that females and males have different mean blood cell counts

B. Construct a 99% confidence interval of the difference between the mean white blood cell count of females and males based on the results listed below. Does there appear to be a difference?

Female: 8.90, 6.50, 9.45 ,7.65, 6.40, 5.15, 16.60, 5.75, 11.60, 5.90, 9:30, 8.55, 10.80,

4.85, 4.90, 8.75, 6.90, 9.75, 4.05, 9.05, 5.05, 6.40, 4.05, 7.60, 4.95, 3.00, 9.10

Male: 5.25, 5.95, 10.05, 5.45, 5.30, 5.55, 6.85, 6.65, 6.30, 6.40, 7.85, 7.70, 5.30, 6.50, 4.55, 7.10, 8.00, 4.70, 4.40, 4.90, 10.75, 11.00 9.60

## Statistics Problem Set: Rejecting the Null Hypothesis

3. Consider the hypothesis test given by

H_0: u = 670

H_1: u does not equal 670

In a random sample of 70 subjects, the sample mean is found to be 678.2. The population standard deviation is known to be omega = 27.

(a) Determine the P-value for this test. (Show work)

(b) Is there sufficient evidence to justify the rejection of H_0 at the omega = 0.02 level? Explain.

4. The playing times of songs are normally distributed. Listed below are the playing times (in seconds) of 10 songs from a random sample. Use a 0.05 significance level to test the claim that the songs are from a population with a standard deviation less than 1 minute.

448 231 246 246 227 213 239 258 255 257

(a) What are your null hypothesis and alternative hypothesis?

(b) What is the test statistic? (Show work)

(c) What is your conclusion? Why? (Show work)

5. Given a sample size of 25, with sample mean 736.2 and sample standard deviation 82.3, we perform the following hypothesis test

H_0: u = 750

H_1: u < 750

What is the conclusion of the test at the omega = 0.10 level? Explain your answer. (Show work)

## Heights of winners and runners-up

Assume that the paired sample data are simple random samples and that the differences have a distribution that is approximately normal.

Heights of Winners and Runners-up

Listed below are the heights (in inches) of candidates who won presidential elections and the heights of the candidates who were runners up. The data are in chronological order, so the corresponding heights from the two lists are matched. For candidates who won more than once, only the heights from the first election are included, and no elections before 1900 are included.

a.) A well-known theory is that winning candidates tend to be taller than the corresponding losing candidates. Use a 0.05 significance level to test that theory. Does height appear to be an important factor in winning the presidency?

b.) If you plan to test the claim in part (a) by using a confidence interval, what confidence level should be used? Construct a confidence interval using that confidence level, then interpret the result.

Won Presidency; – 71, 74.5, 74, 73, 69.5, 71.5, 75, 72, 70.5, 69, 74, 70, 71, 72, 70, 67

Runner-up; – 73, 74, 68, 69.5, 72, 71, 72, 71.5, 70, 68, 71, 72, 70, 72, 72, 72.

## Statistics: Conclusions for Cobalamine (vitamin B12) in Growing Teens

The recommended daily allowance (RDA) of cobalamine (vitamin B12) for growing teens is 2.4ug (micrograms). It is generally believed that growing teens are getting less than the RDA of 2.4ug of cobalamine daily. The FDA managed to collect with a 24-hour period blood samples of 10 randomly selected teens around the country. The amounts of cobalamine (in ug) determined in these 10 randomly selected teens are given as follows:

1.85, 2.35, 1.87, 1.90, 1.37, 2.35, 2.55, 2.28, 1.95, 2.49

Based on their national experience, the FDA assumes that the population standard deviation of cobalamine in teens to be 0.56ug. Now, you are asked to weigh in on the dispute between the FDA and ntbnP.

Perform a hypothesis test. What kind of conclusion can you draw from the hypothesis test? Of course, representatives of ntbnP would like to have the conclusion skewed to their advantage, so would the officials from FDA. What would you do if you are representing ntbnP? If you are representing FDA, how would you present your argument?

## Hypothesis Testing Practice Problems

Please see the attachment for proper formatting and symbols.

7.45 #5. Find a value, X0, such that µ = 160 and o2 = 256

a)P(X < X0) = 0.16

b) P(X-< X0) = 0.75

7.64 #16. A sample of size n = 20 is randomly selected from a normal population with mean µ = 90 and standard deviation = 5. Find the following:-

a. P (x > 95) the x has a bar over the top on all of the problems a-d in this section

b. P (x < 93)

c. P (82 < x < 91)

d. P (x < 89)

9.17 #4. Consider the following hypothesis test:

H0: µ = 5

Ha: µ = > 5

Assume the test statistics are as shown below. Compute the corresponding p-values and make the appropriate conclusions based on ? = .05.

a. Z = 1.82 b. z = .45 c. z = 1.5

9.18 #7. Consider the following hypothesis test.

H0:µ = 25

Ha:µ ?25

A sample of 80 is used and the population standard deviation is 10. Use ? = 0.05. Compute the value of the test statistic z and specify your conclusion for each of the following sample results.

a. X = 22.0

b. X = 24.0

c. X= 23.5

d. X = 22.8

(x has a bar above it on each of the above problems).

## Testing Hypotheses and t Distribution Statistics

3. An earlier study claims that U.S. adults spend an average of 114 minutes with their families per day. A recently taken sample of 25 adults showed that they spend an average of 109 minutes per day with their families. The sample standard deviation is 11 minutes. Assume that the time spent by adults with their families has an approximate normal distribution. We wish to test whether the mean time spent currently by all adults with their families is less than 114 minutes a day.

a) Construct a 95% confidence interval for the mean tine spent by all adults with their families.

b) Does the sample information support that the mean time spent currently by all adults with their families is less than 114 minutes a day? Explain your conclusion in words.

2. A wine manufacturer sells Cabernet Sauvignon with a label that asserts an alcohol content of 11%. Sixteen bottles of this cabernet are randomly selected and analyzed for alcohol content. The resulting observations are:

10.8, 9.6, 9.5, 11.4, 9.8, 9.1, 10.4, 10.7, 10.2, 9.8, 10.4, 11.1, 10.3, 9.8, 9.0, 9.8

The manufacturer claims that the mean alcohol content of its cabernet sauvignon is 11%. At the 98% level of confidence, can we conclude that the claim is true?

3. A shop manual gives 6.5 hours as the average time required to perform a 30,000 mile major maintenance service on a Porsche 911. Last month a mechanic performed 11 such services, and his required times were as follows.

6.3 6.6 6.7 5.9 6.3 6.0 6.5 6.1 6.2 6.4 6.3

At the 95% level of confidence, can we conclude that the mechanic can perform this service in less time than specified by the service manual?

## Statistics: Heights of Presidents and Runners-Up

Please help with the following problem.

Theories have been developed about the heights of winning candidates for the U.S. presidency and the heights of candidates who were runners-up. Listed below are heights (in inches) from recent presidential elections. Is there a linear correlation between the heights of candidates who won and the heights of the candidates who runners-up?

Winner: 69.5, 73, 73, 74, 74.5, 74.5, 71, 71

Runner-up: 72, 69.5, 70, 68, 74, 74, 73, 76

Construct a scatterplot. Find the value of the linear correlation coefficient r and find the critical values of r using alpha = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables.

## Testing a Claim using Significance Testing

Test the given claim – Global warming survey:

A Pew research poll was conducted to investigate opinions about global warming. The respondents who answered yes when asked if there is solid evidence that the earth is getting warmer were then asked to select a cause of global warming. The results for two age brackets are given in the table below. Use a 0.01 significance level to test the claim that the age bracket is independent of the choice for the cause of global warming. Do respondents from both age brackets appear to agree, or is there a substantial difference?

Human activity Natural patterns Don’t know

Under 30 108 41 7

65 and over 121 71 43

## Injuries and Motorcycle Helmet Color

A case-control study was conducted to investigate a relationship between the colors of helmets worn by motorcycle drivers and whether they are injured or killed in a crash. Results are given in the table below. Test the claim that injuries are independent of helmet color. Should motorcycle drivers choose helmets with a particular color? If so, which color appears best?

Color of Helmet

Black White Yellow/Orange Red Blue

Controls (Not Injured) 491 377 31 170 55

Cases (injured or Killed) 213 112 8 70 26.

## Statistics Test Mutual Funds Question

Please help with the following statistics test and mutual funds problems. Step by step calculations are given.

Fidelity Magellan is a large cap growth mutual fund and Fidelity Small Cap Stock is a small cap growth mutual fund (Morningstar Funds 500, 2006). The standard deviation for both funds was computed based on a sample size of 26. For Fidelity Magellan, the sample standard deviation is 8.89%; for Fidelity Small Cap Stock, the sample standard deviation is 13.03%. Financial analysts often use the standard deviation as a measure of risk. Conduct a hypothesis test to determine whether the small cap growth fund is riskier than the large cap growth fund. Use an alpha of 0.05 as the level of significance.

## Research Paper: Major, Age and Gender

Examine the variables in the data file 2004GSS.sav in terms of their labels and values. Develop a research paper with the five sections described below. You should submit your report as one MSWord document with all data and tables copied into that document.

Introduction: The purpose of the paper. Rewrite this section after completing Sections 2-5.

Research Hypotheses: Choose one from the following three variables to be the dependent variable for three alternative hypotheses you will establish:

– Grass

– Fear

– Gunlaw

Choose three other variables in 2004GSS.sav to establish three research hypotheses with the same dependent variable you have selected in Step 4. Each hypothesis should state clearly the direction of the relationship between the pair of variables.

Methods – Secondary Data Analysis: Provide a brief description, in about half a page, of the GSS data in terms of 1) who collected the data, 2) the purpose of the data collection program, 3) data collection method (Experimentation? Self-administered survey? Personal interview? Or existing data?), 4) the study population (i.e., who does the sample represent), and 5) sampling in terms of sample type (e.g., probability/random or none probability/non random?), and 6) sample size. (see 2.2 – 2.5 in the textbook)

Table 1

Descriptive statistics of the variables

Variable Frequency % Mean Std Dev.

Major

Business 253 47.1

Nonbusiness 284 52.9

Gender

Male 253 47.3

Female 282 52.7

Age

Under 30 324 60.6

30 and over 211 39.4

Income $38,620 $17,261

Report descriptive statistics in a table including AGE, RACE, SEX, EDC, INCOME of the respondents. (Hint: Perform descriptive statistics on these variables according to the nature of each variable. For INCOME, you may want to record it into less categories). The following is an example as how to structure the table.

Describe statistical methods you will use to test your three research hypotheses. (Hint: Determine the level of measurement for the variables in each your three hypotheses (in terms of categorical/discrete or continuous/scale).

Findings: Report the results of the observed existence, strength and direction of the relationship (Insert the proper table to where you report the statistics.). (First, perform proper bivariate statistical analysis to test each alternative hypothesis against the H0).

Discussions and conclusion: Do the data bear evidence that support your hypotheses? Any surprises or unexpected results? Your suggestions or recommendations for future studies in terms of data and methods.