Exam 7: Inferences Based on a Single Sample: Estimation With Confidence Intervals

Let $t _ { 0 }$ be a particular value of $t$ . Find a value of $t _ { 0 }$ such that $P \left( t \leq t _ { 0 } \right.$ or $\left. t \geq t _ { 0 } \right) = .1$ where df $= 14$ .

For the given combination of $\alpha$ and degrees of freedom (df), find the value of $\chi _ { \alpha / 2 } ^ { 2 }$ that would be used to find the lower endpoint of a confidence interval for $\sigma ^ { 2 }$ . $\alpha = 0.1 , \mathrm { df } = 14$

A marketing research company is estimating which of two soft drinks college students prefer. A random sample of 148 college students produced the following confidence interval for the Proportion of college students who prefer drink A: (.344, .494). Is this a large enough sample for This analysis to work?

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

The Central Limit Theorem guarantees an approximately normal sampling distribution for the sample mean for large sample sizes, so no knowledge about the distribution of the population is necessary for the corresponding interval to be valid.

For data with two outcomes (success or failure), the binomial proportion of successes is likely to be the parameter of interest.

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). State the appropriate interpretation for This confidence interval. Note that all answers begin with ʺWe are 95 percent confidence that…ʺ

Given the values of $\bar { x } , \mathrm {~s}$ , and $\mathrm { n }$ , form a $99 \%$ confidence interval for $\sigma ^ { 2 }$ . $\overline { \mathrm { x } } = 15.1 , \mathrm {~s} = 4.3 , \mathrm { n } = 24$

A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 95% confidence interval for The mean was calculated to be ($2,181,260, $5,836,180). What would happen to the confidence Interval if the confidence level were changed to 90%?

Sales of a new line of athletic footwear are crucial to the success of a newly formed company. The company wishes to estimate the average weekly sales of the new footwear to within $150 with 99% reliability. The initial sales indicate that the standard deviation of the weekly sales figures is approximately $1550. How many weeks of data must be sampled for the company to get the information it desires?

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

Determine the confidence level for the given confidence interval for $\mu$ . $\bar { x } \pm 1.00 \left( \frac { \sigma } { \sqrt { n } } \right)$

A random sample of 80 observations produced a mean $\bar { x } = 35.4$ and a standard deviation $s$ $= 3.1$ . a. Find a $90 \%$ confidence interval for the population mean $\mu$ . b. Find a $95 \%$ confidence interval for $\mu$ . c. Find a $99 \%$ confidence interval for $\mu$ . d. What happens to the width of a confidence interval as the value of the confidence coefficient is increased while the sample size is held fixed?

Which statement best describes a parameter?

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 $s ^ { 2 } =$ $16 .$

After elections were held, it was desired to estimate the proportion of voters who regretted that they did not vote. How many voters must be sampled in order to estimate the true proportion to within $2 \% ($ e.g., $\pm 0.02$ ) at the $90 \%$ confidence level? Assume that we believe this proportion lies close to $30 \%$ .

The registrarʹs office at State University would like to determine a 95% confidence interval for the mean commute time of its students. A member of the staff randomly chooses a parking lot and Surveys the first 200 students who park in the chosen lot on a given day. The confidence interval is

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 $12,580 with a standard deviation of $700. What is the 90% confidence Interval for the true mean resale value of a 5- year-old car of this model?

A previous random sample of 4000 U.S. citizens yielded 2250 who are in favor of gun control legislation. How many citizens would need to be sampled for a 99% confidence interval to estimate The true proportion within 3%?

What type of car is more popular among college students, American or foreign? One hundred fifty-nine college students were randomly sampled and each was asked which type of car he She prefers. A computer package was used to generate the printout below for the proportion of College students who prefer American automobiles. SAMPLE PROPORTION $= .390998$ SAMPLE SIZE = 159 UPPER LIMIT $= .464240$ LOWER LIMIT $= .331153$ What proportion of the sampled students prefer foreign automobiles?