# STATISTICS

.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?