Exam 6: Inferences Based on a Single Sample: Estimation With Confidence Intervals

You are interested in purchasing a new car. One of the many points you wish to consider is the resale value of the car after 5 years. Since you are particularly interested in a certain foreign sedan, you decide to estimate the resale value of this car with a 90% confidence interval. You manage to obtain data on 17 recently resold 5-year-old foreign sedans of the same model. These 17 cars were resold at an average price of $12,630 with a standard deviation of $800. Suppose that the interval is calculated to be ($12,291.23, $12,968.77). How could we alter the sample size and the confidence coefficient in order to guarantee a decrease in the width of the interval?

For $n = 40 \text { and } \hat { p } = .35 \text {, is }$ s the sample size large enough to construct a confidence for p?

Let $t _ { 0 }$ be a specific value of $t$ . Find $t _ { 0 }$ such that the following statement is true: $P \left( t \geq t _ { 0 } \right) = .05$ where $\mathrm { df } = 20$ .

A random sample of $n = 100$ measurements was selected from a population with unknown mean $\mu$ and standard deviation $\sigma$ . Calculate a $95 \%$ confidence interval if $\bar { x } = 26$ and $s ^ { 2 } = 16$ .

Suppose you want to estimate a population mean $\mu$ and that $\bar { x } = 145 , \mathrm {~s} = 21 , \mathrm { n } = 750$ , and $\mathrm { N } = 4000$ . Find an approximate $95 \%$ confidence interval for $\mu$ .

A sociologist develops a test to measure attitudes towards public transportation, and 47 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.4. Construct an approximate 95% confidence interval for the mean score of all such subjects.

Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is invested in a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of eight private colleges in the United States revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, $122.6,177.5,95.4$ , and 220.0. Summary statistics yield $\bar { x } = 180.975$ and $s = 143.042$ . Calculate a $99 \%$ confidence interval for the mean endowment of all private colleges in the United States.

A newspaper reported on the topics that teenagers most want to discuss with their parents. The findings, the results of a poll, showed that 46% would like more discussion about the family's financial situation, 37% would like to talk about school, and 30% would like to talk about religion. These and other percentages were based on a national sampling of 549 teenagers. Using 99% reliability, can we say that more than 30% of all teenagers want to discuss school with their parents?

For quantitative data, the target parameter is most likely to be the mode of the data.

Suppose you selected a random sample of $n = 29$ measurements from a normal distribution. Compare the standard normal z value with the corresponding $t$ value for a $95 \%$ confidence interval.

Parking at a large university can be extremely difficult at times. One particular university is trying to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. A survey of 338 College of Business (COBA) students yields the following descriptive information regarding the length of time (in minutes) it took them to find a parking spot. Note that the "Lo 95%" and "Up 95%" refer to the endpoints of the desired confidence interval. Variable Lo 95\% CI Mean Up 95\% CI SD Parking Time 338 9.1944 10.466 11.738 11.885 Give a practical interpretation for the 95% confidence interval given above.

Let t0 be a particular value of t. Find a value of $t _ { 0 } \text { such that } P \left( t \leq t _ { 0 } \right) = .005 \text { where } d f = 9 \text {. }$

Suppose that 100 samples of size $n = 50$ are independently chosen from the same population and that each sample is used to construct its own $95 \%$ confidence interval for an unknown population mean $\mu$ . How many of the 100 confidence intervals would you expect to actually contain $\mu$ ?

How much money does the average professional football fan spend on food at a single football game? That question was posed to 60 randomly selected football fans. The sampled results show that the sample mean was $70.00 and prior sampling indicated that the population standard deviation was $17.50. Use this information to create a 95 percent confidence interval for the population mean

Suppose you wanted to estimate a binomial proportion, p, correct to within .03 with probability 0.90. What size sample would need to be selected if p is known to be approximately 0.7?

Sales of a new line of athletic footwear are crucial to the success of a company. The company wishes to estimate the average weekly sales of the new footwear to within $300 with 90% reliability. The initial sales indicate that the standard deviation of the weekly sales figures is approximately $1100. How many weeks of data must be sampled for the company to get the information it desires?

To help consumers assess the risks they are taking, the Food and Drug Administration (FDA) publishes the amount of nicotine found in all commercial brands of cigarettes. A new cigarette has recently been marketed. The FDA tests on this cigarette yielded mean nicotine content of 27.3 Milligrams and standard deviation of 2.1 milligrams for a sample of $n = 9$ cigarettes. Construct a 95% confidence interval for the mean nicotine content of this brand of cigarette.

Find the value of t0 such that the following statement is true: $P \left( - t _ { 0 } \leq t \leq t _ { 0 } \right) = .90 \text { where } \mathrm { df } = 14 \text {. }$

Which information is not shown on the screen below?

A retired statistician was interested in determining the average cost of a $200,000.00 term life insurance policy for a 60-year-old male non-smoker. He randomly sampled 65 subjects (60-year-old male non-smokers) and constructed the following 95 percent confidence interval for the mean cost of the term life insurance: ($850.00, $1050.00). What value of alpha was used to create this confidence interval?