# MATHEMATICS

4.2 Identify the parameter, Part II. For each of the following situations, state whether the

parameter of interest is a mean or a proportion.

(a) A poll shows that 64% of Americans personally worry a great deal about federal spending and the budget deficit.

(b) A survey reports that local TV news has shown a 17% increase in revenue between 2009 and 2011 while newspaper revenues decreased by 6.4% during this time period.

(c) In a survey, high school and college students are asked whether or not they use geolocation

services on their smart phones.

(d) In a survey, internet users are asked whether or not they purchased any Groupon coupons.

(e) In a survey, internet users are asked how many Groupon coupons they purchased over the last year.

4.4 Heights of adults. Researchers studying anthropometry collected body girth measurements

and skeletal diameter measurements, as well as age, weight, height and gender, for 507 physically

active individuals. The histogram below shows the sample distribution of heights in centimeters.

(a) What is the point estimate for the average height of active individuals? What about the

median?

(b) What is the point estimate for the standard deviation of the heights of active individuals?

What about the IQR?

(c) Is a person who is 1m 80cm (180 cm) tall considered unusually tall? And is a person who is

1m 55cm (155cm) considered unusually short? Explain your reasoning.

(d) The researchers take another random sample of physically active individuals. Would you

expect the mean and the standard deviation of this new sample to be the ones given above.

Explain your reasoning.

(e) The samples means obtained are point estimates for the mean height of all active individuals,

if the sample of individuals is equivalent to a simple random sample. What measure do we use

to quantify the variability of such an estimate? Compute this quantity using the data from

the original sample under the condition that the data are a simple random sample.

4.6 Chocolate chip cookies. Students are asked to count the number of chocolate chips in 22

cookies for a class activity. They found that the cookies on average had 14.77 chocolate chips with

a standard deviation of 4.37 chocolate chips.

(a) Based on this information, about how much variability should they expect to see in the mean

number of chocolate chips in random samples of 22 chocolate chip cookies?

(b) The packaging for these cookies claims that there are at least 20 chocolate chips per cookie.

One student thinks this number is unreasonably high since the average they found is much

lower. Another student claims the di_erence might be due to chance. What do you think?

4.8 Mental health. Another question on the General Social Survey introduced in Exercise 4.7

is \For how many days during the past 30 days was your mental health, which includes stress,

depression, and problems with emotions, not good?” Based on responses from 1,151 US residents,

the survey reported a 95% confidence interval of 3.40 to 4.24 days in 2010.

(a) Interpret this interval in context of the data.

(b) What does a 95% confidence level mean in this context?

(c) Suppose the researchers think a 99% confidence level would be more appropriate for this

interval. Will this new interval be smaller or larger than the 95% confidence interval?

(d) If a new survey asking the same questions was to be done with 500 Americans, would the

standard error of the estimate be larger, smaller, or about the same. Assume the standard

deviation has remained constant since 2010.

4.10 Confidence levels. If a higher con_dence level means that we are more confident about

the number we are reporting, why don’t we always report a confidence interval with the highest

possible confidence level?

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.