# MATHEMATICS

4.12 Thanksgiving spending, Part I. The 2009 holiday retail season, which kicked off on November 27, 2009 (the day after Thanksgiving), had been marked by somewhat lower self-reported consumer spending than was seen during the comparable period in 2008. To get an estimate of consumer spending, 436 randomly sampled American adults were surveyed. Daily consumer spending for the six-day period after Thanksgiving, spanning the Black Friday weekend and Cyber Monday, averaged $84.71. A 95% confidence interval based on this sample is ($80.31, $89.11). Determine whether the following statements are true or false, and explain your reasoning.

(a) We are 95% confident that the average spending of these 436 American adults is between

$80.31 and $89.11.

(b) This confidence interval is not valid since the distribution of spending in the sample is right

skewed.

(c) 95% of such random samples would have a sample mean between $80.31 and $89.11.

(d) We are 95% confident that the average spending of all American adults is between $80.31 and

$89.11.

(e) A 90% confidence interval would be narrower than the 95% confidence interval since we don’t

need to be as sure about capturing the parameter.

(f) In order to decrease the margin of error of a 95% confidence interval to a third of what it is

now, we would need to use a sample 3 times larger.

(g) The margin of error for the reported interval is 4.4.

4.14 Age at first marriage, Part I. The National Survey of Family Growth conducted by the

Centers for Disease Control gathers information on family life, marriage and divorce, pregnancy,

infertility, use of contraception, and men’s and women’s health. One of the variables collected on

this survey is the age at first marriage. The histogram below shows the distribution of ages at

first marriage of 5,534 randomly sampled women between 2006 and 2010. The average age at first

marriage among these women is 23.44 with a standard deviation of 4.72

Estimate the average age at _rst marriage of women using a 95% confidence interval, and interpret

this interval in context. Discuss any relevant assumptions.

4.16 Identify hypotheses, Part II. Write the null and alternative hypotheses in words and

using symbols for each of the following situations.

(a) Since 2008, chain restaurants in California have been required to display calorie counts of

each menu item. Prior to menus displaying calorie counts, the average calorie intake of diners

at a restaurant was 1100 calories. After calorie counts started to be displayed on menus, a nutritionist collected data on the number of calories consumed at this restaurant from a random sample of diners. Do these data provide convincing evidence of a difference in the average calorie intake of a diners at this restaurant?

(b) Based on the performance of those who took the GRE exam between July 1, 2004 and June 30, 2007, the average Verbal Reasoning score was calculated to be 462. In 2011 the average verbal score was slightly higher. Do these data provide convincing evidence that the average GRE Verbal Reasoning score has changed since 2004?

4.18 Age at first marriage, Part II. Exercise 4.14 presents the results of a 2006 – 2010 survey showing that the average age of women at first marriage is 23.44. Suppose a researcher believes

that this value has increased in 2012, but he would also be interested if he found a decrease. Below

is how he set up his hypotheses. Indicate any errors you see.

4.20 Thanksgiving spending, Part II. Exercise 4.12 provides a 95% confidence interval for the

average spending by American adults during the six-day period after Thanksgiving 2009: ($80.31,

$89.11).

(a) A local news anchor claims that the average spending during this period in 2009 was $100.

What do you think of this claim?

(b) Would the news anchor’s claim be considered reasonable based on a 90% confidence interval?

Why or why not?

4.22 Gifted children, Part I. Researchers investigating characteristics of gifted children col-

lected data from schools in a large city on a random sample of thirty-six children who were identified as gifted children soon after they reached the age of four. The following histogram shows the distribution of the ages (in months) at which these children first counted to 10 successfully. Also provided are some sample statistics.

(a) Are conditions for inference satisfied?

(b) Suppose you read on a parenting website that children first count to 10 successfully when they

are 32 months old, on average. Perform a hypothesis test to evaluate if these data provide

convincing evidence that the average age at which gifted children first count to 10 successfully

is different than the general average of 32 months. Use a significance level of 0.10.

(c) Interpret the p-value in context of the hypothesis test and the data.

(d) Calculate a 90% confidence interval for the average age at which gifted children first count to

10 successfully. (e) Do your results from the hypothesis test and the confidence interval agree? Explain.

4.24 Gifted children, Part II. Exercise 4.22 describes a study on gifted children. In this study, along with variables on the children, the researchers also collected data on the mother’s and father’s IQ of the 36 randomly sampled gifted children. The histogram below shows the distribution of mother’s IQ. Also provided are some sample statistics.

(a) Perform a hypothesis test to evaluate if these data provide convincing evidence that the average IQ of mothers of gifted children is different than the average IQ for the population at large, which is 100. Use a significance level of 0.10.

(b) Calculate a 90% confidence interval for the average IQ of mothers of gifted children.

(c) Do your results from the hypothesis test and the confidence interval agree? Explain.

4.26 Find the sample mean. You are given the following hypotheses:

We know that the sample standard deviation is 10 and the sample size is 65. For what sample

mean would the p-value be equal to 0.05? Assume that all conditions necessary for inference are

satisfied.

4.28 Testing for food safety. A food safety inspector is called upon to investigate a restaurant with a few customer reports of poor sanitation practices. The food safety inspector uses a hypothesis testing framework to evaluate whether regulations are not being met. If he decides the restaurant is in gross violation, its license to serve food will be revoked.

(a) Write the hypotheses in words.

(b) What is a Type 1 error in this context?

(c) What is a Type 2 error in this context?

(d) Which error is more problematic for the restaurant owner? Why?

(e) Which error is more problematic for the diners? Why?

(f) As a diner, would you prefer that the food safety inspector requires strong evidence or very

strong evidence of health concerns before revoking a restaurant’s license? Explain your reasoning.

4.30 Car insurance savings, Part I. A car insurance company advertises that customers switching to their insurance save, on average, $432 on their yearly premiums. A market researcher at a competing insurance discounter is interested in showing that this value is an overestimate

so he can provide evidence to government regulators that the company is falsely advertising their

prices. He randomly samples 82 customers who recently switched to this insurance and finds an

average savings of $395, with a standard deviation of $102.

(a) Are conditions for inference satisfied?

(b) Perform a hypothesis test and state your conclusion.

(c) Do you agree with the market researcher that the amount of savings advertised is an overestimate? Explain your reasoning.

(d) Calculate a 90% confidence interval for the average amount of savings of all customers who

switch their insurance.

(e) Do your results from the hypothesis test and the confidence interval agree? Explain.

4.32 Speed reading, Part I. A company offering online speed reading courses claims that students who take their courses show a 5 times (500%) increase in the number of words they can read in a minute without losing comprehension. A random sample of 100 students yielded an average increase of 415% with a standard deviation of 220%. Is there evidence that the company’s claim is false?

(a) Are conditions for inference satisfied?

(b) Perform a hypothesis test evaluating if the company’s claim is reasonable or if the true average improvement is less than 500%. Make sure to interpret your response in context of the hypothesis test and the data. Use α= 0:025.

(c) Calculate a 95% confidence interval for the average increase in the number of words students

can read in a minute without losing comprehension.

(d) Do your results from the hypothesis test and the confidence interval agree? Explain.