Suppose that researchers study a sample of 50 people and find that 10 are left-handed.

(a) Find a 95% confidence interval for the population proportion that is left-handed.

(b) What would the confidence interval be if the researchers used the Wilson value ~p instead?

(c) Suppose that an investigator tests the null hypothesis that the population proportion is 18% against the alternative that it is less than that. If = 0:05 then nd the critical value ^pc. Using ^p as the sample estimate, would the investigator reject the null?

(d) Suppose that researchers are using this critical value but, unbeknownst to them, the true, population proportion is 0.16. Find the power of the test.

Data has been collected on the number of hospital admissions resulting from car accidents for Fridays on the 6^th of the month and Fridays on the following 13^th of the same month. Test the superstition that there is a difference between the mean admissions on the two days. Let alpha be .05.

6^th                    13^th

9                      13

6                      12

11                    14

11                    10

3                      4

5                      12

A vegetable distributor knows that during the month of August, the weights of its tomatoes are normally distributed with a mean of 0.52 lb and a standard deviation of 0.15 lb. (See Example 2 in this section.)

(a) What percent of the tomatoes weigh less than 0.67 lb? %

(b) In a shipment of 6,000 tomatoes, how many tomatoes can be expected to weigh more than 0.22 lb? tomatoes

(c) In a shipment of 3,500 tomatoes, how many tomatoes can be expected to weigh from 0.22 lb to 0.82 lb?

You would like to determine if teacher A gives different grades from teacher B in the same class at a significance level of 0.01. You take a random sample of 85 students from teacher A, 61 of whom passed the class. You also take a random sample of 70 students from teacher B, 56 of whom passed the class.

a) What is the estimate of the difference in proportion of students who pass? Find the estimate of pA − pB.

b) Is it appropriate to assume that the sampling distribution of the difference in proportions is normally distributed?

c) Find a 90% confidence interval for the difference in part (a) and interpret it.

d) Your original goal is to test to see if the two teachers are different. What are the hypotheses for this test?

e) Calculate the z-test statistic for the tests defined in part (d). f) Find the P-value for the test defined in part (d). g) State your conclusion for the hypothesis test, and then interpret the result in the context of the problem.

Standard deviation is a useful concept in performance management. Let us say that a director in a local fire department wants to know any variation between the performance of this year and that of the last year. He draws a sample of 10 response times of this year ( in minutes): 3.0, 12.0, 7.0, 4.0, 4.0, 6.0, 3.0, 9.0, 11.0, and 15.0, comparing them with a sample of 10 response times last year ( in minutes): 8.0, 7.0, 8.0, 6.0, 6.0, 9.0, 7.0, 9.0, 8.0, and 6.0.

a. Does he see a performance variation by the mean? ( 10 points)

b. Does he see a performance variation by the standard deviation? If he does, is it performance improvement or deterioration from the last year? Why? ( 10 points)

The management of a supermarket wanted to investigate whether the male customers spend less money on average than female customers. A sample of 22 male customers who shopped at this supermarket showed that they spent an average of $73 with a standard deviation of $17.50. Another sample of 20 female customers who shopped at the same supermarket showed that they spent an average of $87 with a standard deviation of $14.40. Assume that the amounts spent at this supermarket by all male and female customers are normally distributed with unequal and unknown population standard deviations.

a.) Construct a 90% confidence interval for the difference between the mean amounts spent by all male and all female customers at this supermarket.

b.) Using the 2.5% significance level, can you conclude that the mean amount spent by all male customers at this supermarket is less than that by all female customers?

c) Interpret the conclusion obtained in part b.

The Enormous State University Public Relations Department has issued a statement claiming that 80% of those

who enroll at ESU graduate within 6 years. Being naturally skeptical, you decide to test their claim. You

interview 500 randomly selected people from the population that enrolled at ESU just over 6 years ago. Of

those 500, 360 have graduated.

a. What is the variable you are analyzing here? Of what sort is it? Who/what are the subjects?

b. What statistic can you compute from the sample?

c. Carefully write out the hypotheses for your test.

d. Run the test! Remember, if you use a normal approximation, you must justify it!

e. Give a “plain English” conclusion for your test.

f. If needed, give a 95% confidence interval for p, with an explanation. Remember the difference between CI and hypothesis testing calculations for

Find the optimum strategies for player A and player B in the game represented by the following payoff matrix. Find the value of the game.

1. A standardized college admissions test has a population mean of 65, a population standard deviation of 10 points, and is normally distributed. If a student scores a 73 on the test, above what percentage of the test-takers does her score fall? Be sure to show your calculations.

Use the given degree of confidence and sample data to construct a confidence interval for the population mean mu.

Assume that the population has a normal distribution. Round to two decimal places. nequals​10, x overbarequals14.5​, sequals4.7​, ​95% confidence

A. 11.78less thanmuless than17.23

B. 11.14less thanmuless than17.86

C. 11.19less thanmuless than17.81

D. 11.15less thanmuless than17.85

Assignment 2: Connection between Confidence Intervals and Sampling Distributions:

The purpose of this activity is to help give you a better understanding of the underlying reasoning behind the interpretation of confidence intervals. In particular, you will gain a deeper understanding of why we say that we are “95% confidentthat the population mean is covered by the interval.”

When the simulation loads you will see a normal-shaped distribution, which represents the sampling distribution of the mean (x-bar) for random samples of a particular fixed sample size, from a population with a fixed standard deviation of σ.

The green line marks the value of the population mean, μ.

To begin the simulation, click the very top “sample” button at the topmost right of the simulation. You will see a line segment appear underneath the distribution; you should see that the line segment has a tiny red dot in the middle.

You have used the simulation to select a single sample from the population; the simulation has automatically computed the mean (x-bar) of your sample; your x-bar value is represented by the little red dot in the middle of the line segment. The line segment represents a confidence interval. Notice that, by default, the simulation used a 95% confidence level.

Question 1:

Did your 95% confidence interval contain (or “cover”) the population mean μ (the green line)?

If your confidence interval did cover the population mean μ, then the simulation will have recorded 1 “hit” on the right side of the simulation.

Now, click to select another single sample.

Question 2:

Was your second sample mean x-bar (the new red dot) the same value as your 1st sample mean? (i.e., is it in the same relative location along the axis?) Why is this result to be expected?

Question 3:

A new 95% confidence interval has also been constructed (the new line segment, centered at the location of your second x-bar). Does the new interval cover the population mean μ?

Notice, under “total” on the right side of the simulation, the number of total selected samples has been tallied.

Now click “sample 50” repeatedly until the simulation tallies a “total” of around 1,000 samples. You will see that the simulation computes the “percent hit” for all the intervals.

Question 4:

What percentage of the many 95% confidence intervals should cover the population mean μ?

Question 5:

Now let’s summarize some key ideas.

Based on what you’ve seen on the simulation (with the level set at 95%), decide which of the following statements are true and which are false.

1. Each interval is centered at the population mean (μ).
2. Each interval is centered at the sample mean (x-bar).
3. The population mean (μ) changes when different samples are selected.
4. The sample mean (x-bar) changes when different samples are selected.
5. In the long run, 95% of the intervals will contain (or “cover”) the sample mean (x-bar).
6. In the long run, 95% of the intervals will contain (or “cover”) the population mean (μ).

A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8,000. A survey of owners of that tire design is conducted. Of the 28 tires surveyed, the mean lifespan was 46,500 miles with a standard deviation of 9,800 miles. Using alpha = 0.05, is the data highly consistent with the claim? Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) A.) State the distribution to use for the test. (Round your answers to two decimal places.) B.) What is the test statistic? (If using the z distribution round your answers to two decimal places, and if using the t distribution round your answers to three decimal places.) C.) What is the p-value? (Round your answer to four decimal places.) D.) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha (Enter an exact number as an integer, fraction, or decimal.) E.) Construct a 95% confidence interval for the true mean. Sketch the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. (Round your lower and upper bounds to the nearest whole number.)

The mean number of sick days an employee takes per year is believed to be about 10. Members of a personnel department do not believe this figure. They randomly survey 8 employees. The number of sick days they took for the past year are as follows: 12; 6; 15; 5; 11; 10; 6; 8. Let X = the number of sick days they took for the past year. Should the personnel team believe that the mean number is about 10? Conduct a hypothesis test at the 5% level. Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) A.) State the distribution to use for the test. (Enter your answer in the form z or tdf where df is the degrees of freedom.) B.) What is the test statistic? (If using the z distribution round your answers to two decimal places, and if using the t distribution round your answers to three decimal places.) C.) What is the p-value? (Round your answer to four decimal places.) D.) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha (Enter an exact number as an integer, fraction, or decimal.) E.) Construct a 95% confidence interval for the true mean. Sketch the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. (Round your answers to three decimal places.)

Farmer's Corporation made a report that the mean annual household income of its readers is $119,155 (Thebill, March 2017). Assume this estimate of the mean annual household income is based on a sample of 80 households, and, based on past studies the population standard deviation is known to be σ = $30,000.00.

1. Developed a 90% confidence interval estimate of the population mean

2. Developed a 95% confidence interval estimate of the population mean.

3. Developed a 99% confidence interval estimate of the population mean

4. Discuss what happens to the width of the confidence interval as the confidence level is increased. Does this result seem reasonable? Explain.