# Probability, Confidence Interval & Hypothesis Testing

Note: Please round all probability answers correct to three places.

1. A university found that 22% of its students who registered for Statistics I withdraw without completing the course. Assume that 25 students register for Statistics I.

a. Compute the probability that exactly 8 students will withdraw without completing the course.

b. Compute the probability that fewer than 6 students will withdraw without completing the course.

c. Compute the probability that more than 5 students will withdraw without completing the course.

d. Compute the probability that no more than 6 students will withdraw without completing the course.

e. Compute the probability that between 3 and 10 students will withdraw without completing the course

f. Compute the expected number of students who will withdraw without completing the course.

2. The amount of time it takes Susie to make a pie is approximately normally distributed, with a mean of 32 minutes and a standard deviation of 4 minutes.

a. Compute the probability that it takes Susie more than 40 minutes to make a pie.

b. Compute the probability that it takes Susie between 28 and 38 minutes to make a pie.

c. Compute the probability that one day it takes Susie less than 25 minutes to make a pie.

d. Compute the number of minutes that represent the 35th percentile ( P35).

3. The time it takes a seamstress to measure, cut, and hem a wedding dress is continuous and uniformly distributed between the values of 42 minutes and 57 minutes. Determine the following:

a. Draw the graph of this continuous uniform distribution function. (Label both axes and includes proper scales!)

b. What is the probability that it takes between 45 and 49 minutes for a seamstress to finish the hem?

c. What is the probability that it takes more than 47 minutes for a seamstress to finish the hem?

d. What is the probability that it takes 48 minutes exactly for a seamstress to finish the hem?

e. What is the mean time (expected value) for a seamstress to finish the hem?

4. The mean hourly pay rate for employees of fast food restaurants is normally distributed with a mean of $8.50 and a standard deviation of 1.25. If a simple random sample of 25 fast food employees is chosen:

a. What is the probability that the mean of the sample, (),will be between $8.00 and $8.75?

b. What is the probability that the mean of the sample, (), will be $8.00 or less?

c. What is the mean of this distribution? What is the standard deviation of this distribution? (Use the correct symbols to represent these values.)

5. A farmer wants to estimate the mean time it will take for his corn stalks to produce a mature ear of corn. He takes a sample of 12 stalks and records the number of days between planting and maturing. The data, in number of days, is: 62 68 59 85 73 78 61 82 75 79 80 71

a. What is the point estimate for the mean number of days that it takes his corn to mature?

b. Compute the 95% confidence interval of the number of days the corn takes to mature.

c. What is the margin of error in the confidence interval?

d. Interpret the confidence interval from above.

6. A local hairdresser wants to estimate the percent of customers who cancel, don't show up, or reschedule appointments. She wants to be 90% certain of her estimate, and wants to be within 2% of the actual value.

a. Compute the sample size required if she thinks that the percent is possibly 35%.

b. Compute the sample size required if she has no idea what the percent will be.

c. In what two ways could the hairdresser decrease the sample size needed?

1. _______________________________________________

2. _______________________________________________

7. Speedy Rooter claims that it can unclog any drain in 15 minutes or less. If it takes longer than 15 minutes, the customer doesn't pay for the service. The owner of Speedy Rooter wants to know if his claim is really valid. He takes a random sample of 35 drain jobs and finds that the mean time is 15.4 minutes. From past experience, it is known that the population standard deviation, σ, is about 1.4 minutes. Test the company's claim that any drain can be unclogged in 15 minutes or less. Use a level of significance of .05. Use the p-value approach, but be sure to show all steps - hypothesis, graph, p-value, decision, and statement about the problem.

8. A random sample of 49 lunch customers was taken at a restaurant. The average amount of time the customers in the sample stayed in the restaurant was 33 minutes. From past experience, it is known that the standard deviation is 10 minutes. Construct a 95% confidence interval for the true average amount of time customers spent in the restaurant. (Show calculator command or formula)

9. A sheriff read a report stating that nationally 28% of cars did not come to a complete stop before progressing through a stop sign. He wanted to know how the percentage in his town compared to the national average. He asked for data from fellow officers in the town, and randomly assigned them times and locations. He received reports of 850 cars passing through stop signs, with 265 of them not completely stopping. Using a .01 level of significance, use a hypothesis test to see if the sheriff's percentage of cars that do not stop completely is different from the national average for cars not stopping at the stop signs. Use the traditional method. (List the hypotheses, draw and shade the graph, label the critical value, compute the test statistic, make your decision, and make a statement about the problem.)

10. Executives of a coal company want to determine a 95% confidence interval estimate for the average daily tonnage of coal that they mine. Assuming that the company reports state that the standard deviation of daily output is 200 tons, how many days should they sample so that the margin of error will be 39.2 tons?

See attached.

© BrainMass Inc. brainmass.com October 25, 2018, 4:27 am ad1c9bdddfhttps://brainmass.com/math/probability/probability-confidence-interval-hypothesis-testing-379142

#### Solution Summary

The solution provides step by step method for the calculation of probabilities, confidence intervals and testing of hypothesis. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

Statistics: probability, confidence interval, mean, standard deviation, test statistic

See attached four problems.

Please show all work and please follow all instructions.

1. Avery short quiz has one multiple choice questions with five possible choices (a, b, c, d, e) and one true or false question. Assume you are taking the quiz but do not have any idea what the correct answer is to either question, but you mark an answer any way.

a. What is the probability that you have given the correct answer to both questions?

b. What is the probability that only one of the two answers is correct?

c. What is the probability that neither answer is correct?

d. What is the probability that only your answer to the multiple choice question is correct?

e. What is the probability that you have only answered the true or false question correctly?

2. Information regarding the price of a roll of camera film (35 mm, 24 exposure) for a sample of 12 cities world wide is shown below. Determine 91% confidence interval for the population mean.

Price of film information is given in the attachment.

3. Confirmed cases of West Nile virus in birds for a sample of six countries in the state of Georgia are shown below.

Country Cases

Catoosa 6

Chattoogan 3

Dade 3

Gordon 5

Murray 3

Walker 4

You want to determine if the average number of cases of West Nile virus in the state of Georgia is significantly more than 3. Assume the population is normally distributed.

a. State the null and alternative hypotheses

b. Compute the mean and the standard deviation of the sample.

c. Compute the standard error of the mean.

d. Determine the test statistic.

e. Determine the P-value and 95% confidence, test the hypotheses.

4. Consider the following hypotheses test.

H0: mu >= 80

Ha: mu < 80

A sample of 121 provided a sample mean of 77.3. The population statndard deviation is known to be 16.5.

a. Compute the value of the test statistic.

b. Determine the P-value; and at 93.7% confidence, test the above hypotheses.

c. Using the critical value approach at 93.7% confidence, test the hypotheses.

5. In the last Presidential election, a national survey company claimed that no more than 50% (i.e., <=50%) of all registered voters voted for the Republican candidate. In a random sample of 400 registered voters, 208 voted for the Republican candidate.

a. State the null and alternative hypotheses

b. Compute the test statistic.

c. At 95% confidence, compute the P-value and test the hypotheses.

6. Consider the following hypothesis test:

H0: mu <= 38

Ha: mu > 38

You are given the following information obtained from a random sample of six observations. Assume the population has a normal distribution.

X

38

40

42

32

46

42

a. Compute the mean of the sample.

b. Determine the standard deviation of the sample.

c. Determine the standard error of the mean.

d. Compute the value of the test statistic.

e. At 95% confidence using the P-value approach, test the above hypotheses.

7. A test on world history was given to a group of individuals before and also after a film on the history of the world was presented. The results are given in the attachment. We want to determine if the film significantly increased the test scores.

a. Give the hypotheses for this problem.

b. Compute the test statistic.

c. At 95% confidence, test the hypotheses.

8. The Dean of students at UTC has said that the average grade of UTC students is higher than that of the students at GSU. Random samples of grades from the two schools are selected, and the results are shown in the attachment.

a. Give the hypotheses.

b. Compute the degrees of freedom for this test.

c. Compute the test statistic.

d. At a 0.1 level of significance, test the Dean of Student's statement.