Errors and Their Implications

It has been said that from the point of view of the general population, Type I errors are particularly undesirable. This has been used as a justification for setting alpha at such low levels as .05 and .01.

Does this make sense? Explain your response.
What are the implications of making a Type I error, as compared to those of making a Type II error?
Describe the ethical issues in this trade-off.

Random Sampling Distribution Test

An automobile manufacturer advertises that the mean petrol consumption of its new hybrid car does not exceed 4.6 litres per 100 km. An independent company decides to test the manufacturer’s claim by selecting a random sample of 16 vehicles and measuring the consumption of each. They have found that the petrol consumptions in litres per 100 km of those vehicles are as follows:
4.7 4.9 4.4 4.5 4.6 4.7 4.7 4.8 4.9 4.3 5.0 4.6 4.7 4.9 4.8 5.1

a) Perform a hypothesis test at the 5% level to see if there is evidence to support the manufacturer’s claim.

b) State the distributional assumption required for the test in part (a) to be valid, and use an appropriate plot to test the assumption. With reference to your plot, briefly explain why the assumption is or is not satisfied.

c) Based on your answers to parts a) and b), should the manufacturer be reported to a fair trading commission for misleading advertising? Briefly explain your decision.

Uniform distribution of bicycle accidents

A bicycle safety organization claims that fatal bicycle accidents are uniformly distributed throughout the week. The table below shows the day of the week for which 778 randomly selected fatal bicycle accidents occurred. At ? = 0.10, can you reject the claim that the distribution is uniform? Complete parts (a) through (d) below.
(a) State H? and H? and identify the claim
(b) Determine the critical value and the rejection region
(c) Calculate the text statistic
(d) Decide whether to reject or fail to reject the null hypothesis. Then interpret the decision in the context of the original claim. Choose the correct highlighted options.

T-Statistics

Numeric results for several independent-samples t tests are presented here. Decide whether each test is statistically significant, and report each result in the standard APA format.

a. A total of 73 people were studied, 40 in one group and 33 in the other group. The test statistic was calculated as 2.126 for a two-tailed test with a p level of 0.05.

b. One group of 23 people was compared to another group of 18 people. The t statistic obtained for their data was 1.77. Assume you were performing a two-tailed test with a p level of 0.05.

c. One group of 9 mice was compared to another group of 6 mice, using a two-tailed test at a p-level of 0.01. The test statistic was calculated as 3.02.

Creating Hypotheses in A Paint Manufacturing Company

Assume you are the manager of a paint manufacturing factory. Your company has received complaints from customers that the containers hold less than the amount printed on them. On the other hand, corporate management is concerned that the containers hold more than the standard amount. You assign a statistician to verify these claims. A sample of containers was selected and the volume of paint in each container was measured. Assuming that the volume printed on each container is 1 gallon, how would you formulate the null and alternative hypotheses to test the customers’ claim? As a manager, what reasonable criteria will you use to set a value for the level of significance to be used in the test? After answering this question, what type of error would you suppose may result in that case?

Choosing a Suitable Hypothesis Test

For each of the following three scenarios, state which hypothesis test you would use from among the four introduced so far: the z test, the single-sample t test, the paired-samples t test, and the independent-samples t test. (Note: In the actual studies described, the researchers did not always use one of these tests, often because the actual experiment had additional variables.) Explain your answer.

a. A study of children who had survived a brain tumor revealed that they were more likely to have behavioral and emotional difficulties than were children who had not experienced such a trauma (Upton & Eiser, 2006). Forty families participated in the study. Parents rated children’s difficulties, and the ratings data were compared with known means from published population norms.

b. Talarico and Rubin (2003) recorded the memories of 54 students just after the terrorist attacks in the United States on September 11, 2001: some memories related to the terrorist attacks on that day (called flashbulb memories for their vividness and emotional content) and some everyday memories. They found that flashbulb memories were no more consistent over time than everyday memories, even though they were perceived to be more accurate.

c. The HOPE VI Panel Study (Popkin & Woodley, 2002) was initiated to test a U.S. program aimed at improving troubled public housing developments. Residents of five HOPE VI developments were examined at the beginning of the study so researchers could later ascertain whether their quality of life had improved. Means at the beginning of the study were compared to known national data sources (e.g., the U.S. Census, the American Housing Survey) that had summary statistics, including means and standard deviations.

Hypothesis Test: Fitness Camp Glasses

Researchers at the Cornell University Food and Brand Lab conducted an experiment at a fitness camp for adolescents (Wansink & van Ittersum, 2003). Campers were given either a 22-ounce glass that was tall and thin or a 22-ounce glass that was short and wide. Campers with the short glasses tended to pour more soda, milk, or juice than campers with the tall glasses.

a. Is it likely that the researchers used random selection? Explain.
b. Is it likely that the researchers used random assignment? Explain.
c. What is the independent variable, and what are its levels?
d. What is the dependent variable?
e. What hypothesis test would the researchers use?
f. Conduct step 1 of hypothesis testing.
g. Conduct step 2 of hypothesis testing.
h. How could the researchers redesign this study so that they could use a paired-samples t test?

Testing Research Hypotheses

Please help with the following problem.

In playing the Lemonade Stand Game, Bob decreased his price per cup by two cents, from $.27 per cup to $.25 per cup. At the .05 level of significance, did net revenue increase? The data for weeks 3 and 4 are presented below; there were 30 observations in each week.

Items to compare:

Price per Xup
Week 3 ; Week 4
.25 ; .27

Stand. Del
3.77 ; 4.52

Ave. Daily Net Revenue
$2.25 ; $2.10

Describe the steps in testing a research hypothesis. What formula should we use to make the calculation? Make the calculation. What is your decision regarding the null hypothesis?

30 Insulators Level of Significance: Evaluating Force

An organization thinks that if the electrical insulators break when in use, a short circuit might occur. To test the strength of the insulators, a hard core test will be performed to determine how much force is required to break the insulators. Force is measured by observing the number of pounds of force (lbs.) applied to the insulators before it has a chance to break. The data from 30 insulators subjected to this testing is tabulated below (please refer to attached file to view the chart for this question).

Questions:
a. At the 0.05 level of significance, is there evidence that the population mean force is greater than 1,500 lbs.?
b. What assumption about the population distribution is needed in order to conduct the t test in (a)?
c. Construct a histogram and boxplot to evaluate the assumption made in (b).
d. Do you think that the assumption needed in order to conduct t test in (a) is valid? Explain.

Critiquing PERT

You decide to introduce the concept of PERT to a business who is struggling with project scheduling. One of the managers of the company tells you the following:

“Well, we thought about PERT before actually. The best part about PERT is that it recognizes uncertainty in project time estimation as it is – there is no attempt to sweep it below the carpet, and pretend that it does not exist. On the contrary, PERT gives a rough approximation of the uncertainty in the final completion time. Of course, the simplifying assumptions made during the analysis do lead to an underestimation of the probability of long project completion times.

But, as with everything else in life, PERT too is not without its problems. While going through the process of analyzing project and activity completion times, you have to make quite a few simplistic assumptions. We have to assume, for example, that the critical path does not change. However, it is possible that the critical path that was identified based on the most likely or expected completion time will not necessarily end up being the critical path. Scenarios in which another path takes longer than the identified critical path may be ignored. There might be a tendency to understate the expected completion time, and this will definitely underestimate the probability of late completion. And that is why we don’t consider using PERT.”

Critique the manager’s claim. (Do you agree or disagree with the manager’s claim? Why or why not? Can you give any examples of how he is right/wrong? If you believe that underestimation can happen, is it a big problem? Do you think that a non-changing critical path is bad? Why or why not?)

Statistics Problem Set: Hypothesis Testing

1. What is the difference in average daily hotel room rates between Minneapolis and New Orleans? Suppose we want to estimate this difference by taking hotel rate samples from each city and using a 98% confidence level. The data for such a study follow. Use these data to produce a point estimate for the mean difference in the hotel rates for the two cities. Assume the population variances are approximately equal and hotel rates in any given city are normally distributed

Minneapolis
n = 22
average = $112
s = $11

New Orleans
n = 20
average = $122
s = $12
2. Use the data given to test the following hypotheses (? = .05). Assume the differences are normally distributed in the population.

Ho: D = 0 Ha: D does not equal 0

Individual Before After
1 107 102
2 99 98
3 110 100
4 113 108
5 96 89
6 98 101
7 100 99
8 102 102
9 107 105
10 109 110
11 104 102
12 99 96
13 101 100

3. Does age make a difference in the amount of savings a workers feels is needed to be secure at retirement? A study by CommSciences for Transamerica Asset Management found that .24 of workers in the 25-33 age category felt that $250,000-$500,000 is enough to be secure at retirement. However, .35 of the workers in the 34-52 age category feel that this amount is enough. Suppose 210 workers in the 25-33 age category and 176 workers in the 34-52 age category were involved in this study. Use these data to construct a 90% confidence interval to estimate the difference in population proportions on this question.

4. How long are resale houses on the market? One survey by the Houston Association of Realtors reported that in Houston, resale houses are on the market an average of 112 days. Of course, the length of time varies by market. Suppose random samples of 13 houses in Houston and 11 houses in Chicago that are for resale are traced. The data shown here represent the number of days each house was on the market before being sold. Use the given data and a 1% level of significance to determine whether the population variances for the number of days until resale are different in Houston than in Chicago. Assume the number of days resale houses are on the market are normally distributed.

Houston
132 126
138 94
131 161
127 133
99 119
126 88
134

Chicago
118 56
85 69
113 67
81 54
94 137
93.

Statistics Problem Set: Hypothesis Testing Using Samples

Problem 1:
A study by Hewitt Associates showed that 79% of companies offer employees flexible scheduling. Suppose a researcher believes that in accounting firms this figure is lower. The researcher randomly selects 415 accounting firms and through interviews determines that 303 of these firms have flexible scheduling. With a 1% level of significance, does the test show enough evidence to conclude that a significatly lower proportion of accounting firms offer employees flexible scheduling.
Problem 2:
A previous experience shows the variance of a given process to be 14. Researchers are testing to determine whether this value has changed. They gather the following dozen measurements of the process. Use these data and a significance level of .05 to test the null hypothesis about the variance. Assume the measurements are normally distributed.

52 44 51 58 48 49
38 49 50 42 55 51

Problem 3:
Highway engineers in Ohio are painting white stripes on a highway. The stripes are supposed to be approximately 10 feet long. However, because of the machine, the operator, and the motion of the vehicle carrying the equipment, considerable variation occurs among the stripe lengths. Engineers claim that the variance of stripes is not more than 16 inches. Use the sample lengths given here from 12 measured stripes to test the variance claim. Assume stripe length is normally distributed. Let the significance level = .05

Stripe Lengths in Feet
10.3 9.4 9.8 10.1
9.2 10.4 10.7 9.9
9.3 9.8 10.5 10.4.

Statistics: Using the p-value method

Question: Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical conclusion about the null hypothesis and final conclusion that addresses the original claim. Use the p-value method and use the normal distribution as an approximation to the binomial distribution.

Gender selection for boys
The Genetics and IVF Institute conducted a clinical trial of the YSORT method designed to increase the probably that a baby is a boy. As of this writing, among the babies born to parents using the YSORT method, 172 were boys and 39 were girls. Use the sample data with a 0.01 significance level to test the claim that with this method, the probability of a baby being a boy is greater than 0.5. Does the YSORT method of gender selection appear to work?

Statistics: Evaluating significance using the p-value method

Identify the null hypothesis, alternative hypothesis, test statistic, p-value or critical value, and the conclusion about the null hypothesis. Use the P-value method.

Question: When a fair die is rolled many times, the outcomes of 1, 2, 3, 4, 5, and 6 are equally likely, so the mean of the outcomes should be 3.5. The author drilled holes into the die and loaded it by inserting lead weights, then rolled it 16 times to obtain a mean of 2.9375. Assume that the standard deviation of the outcome is 1.7078, which is the standard deviation for a fair die. Use a 0.05 significance level to test the claim that outcomes from the loaded die have a mean different from the value of 3.5 expected with a fair die. Is there anything about the sample data suggesting that the methods of this section should not be used?

Examining raw data

Identify the null hypothesis, alternative hypothesis, test statistic, P-value and conclusion about the null hypothesis. Use the P-value method.

California Speeding.
Listed below are recorded speeds (in mi/h) of randomly selected cars traveling on a section of Highway 405 in Los Angles. That part of the highway has a posted speed limit of 65 mi/h. Assume that the standard deviation of speed is 5.7 mi/h and use a 0.01 significance level to test the claim that the sample is from a population with a mean that is greater than 65 mi/h.

68, 68, 72, 73, 65, 74, 73, 72, 68, 65, 65, 73, 66, 71, 68, 74, 66, 71, 65, 73,
59, 75, 70, 56, 66, 75, 68, 75, 62, 72, 60, 73, 61, 75, 58, 74, 60, 73, 58, 75

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