17. Claim: The average cost to repair washing machine A is the same as the average cost to repair washing machine B. Test at α = 0.05 Data: A sample of 14 washing machine A’s have an average repair cost of $208 and a standard deviation of $25. A sample of 16 washing machine B’s has an average repair cost of $221 and a standard deviation of $29.
Assume that the population standard deviations for repair costs are the same for each
(a) Are these data statistically significant evidence to support the claim?

(b) Are these data statistically significant evidence to refute the claim?

A U.S. Senate Judiciary Committee report showed the number of homicides in each state. In Indiana, Ohio, and Kentucky, the number of homicides was, respectively, 380, 760, and 260. Suppose a stratified random sample with the following results was taken to learn more about the victims and the cause of death.

a. Develop an approximate 95% confidence interval for the proportion of shooting deaths in Indiana (to 4 decimals).

b. Develop an estimate for the total number of shooting deaths in Ohio (to whole number).

c. Develop an approximate 95% confidence interval for the proportion of shooting deaths in Ohio (to 4 decimals).

d. Develop an approximate 95% confidence interval for the proportion of shooting deaths across all three states (to 4 decimals).

A recent graduate moving to a new job collected a sample of the monthly rent (in dollars) and the size (in square feet) of 3-bedroom apartments in one area of a city. Use Excel to find the best fit linear regression equation, where size of the apartment is the explanatory variable. Round the slope and intercept to two decimal places.

Size   Rent
961   "909 "
949   1048
924   "841 "
920   "704 "
902   1131
893   "867 "
888   2532
886   "800 "
876   1030
875   "758 "
864   "930 "
860   1227
851   "931 "
848   1820
845   "711 "
841   1040
838   1373
829   1158
827   3111
815   "733 "
802   "870 "
801   1464
792   "839 "
790   1387
779   2050
748   3275
741   1996
730   1708
727   4031
711   1756

An important quality characteristic used by the manufacturer of Boston and Vermont asphalt shingles is the amount of moisture the shingles contain when they are packaged. Customers may feel that they have purchased a product lacking in quality if they find moisture and wet shingles inside the packaging. In some cases, excessive moisture can cause the granules attached to the shingles for texture and coloring purposes to fall off the shingles, resulting in appearance problems. To monitor the amount of moisture present, the company conducts moisture tests. A shingle is weighed and then dried. The shingle is then re-weighed, and, based on the amount of moisture taken out of the product, the pounds of moisture per 100 square feet are calculated. The company would like to show that the mean moisture content is less than 0.35 pound per 100 square feet. The file Moisture includes 36 measurements ( in pounds per 100 square feet) for Boston shingles and 31 for Vermont shingles.

e. What assumption about the population distribution is needed in order to conduct the t tests in ( a) and ( c)?

f. Construct histograms, boxplots, or normal probability plots to evaluate the assumption made in ( a) and ( c).

g. Do you think that the assumption needed in order to con-duct the t tests in ( a) and ( c) is valid? Explain.

Complete the following problems using R. PLEASE SHOW ANSWER IN R FORMAT

and be sure to turn in explanations and interpretations where appropriate.

2.A random sample of 15 values of PAR (photosynthetically active radiation) were taken at noon in a certain forest (in moles per meter²per second). Data are included in the assignment .xlsx file. Assume that PAR is approximately normally distributed with a known standard deviation of 40.71 moles/(m²s).

a.What is the mean value observed for PAR?

b.What is the standard error of the mean value of PAR, given this sampling scenario?

c.Find the 95% confidence intervals for the unknown population mean of the PAR values and interpret its meaning.  (Note: for full credit, you must write down a probability statement about the unknown population mean.)

d.Find the 90% confidence intervals for the unknown population mean of the PAR values and interpret its meaning.

e.Compare the two intervals.

f.Test the hypothesis that average PAR value is 550 moles/(m²s), at the =0.05 level. (Be sure to right down all steps – including hypotheses – as in the lecture notes, and interpret the meaning of the test!)

Book: Probability and Statistical Inference (9th & 10th edition respectively)

Chapter 8: Tests of Statistical Hypotheses

Section 8.6 [Best Critical Regions]: Problem 8 (9th Edition) OR Section 8.7 [Best Critical Regions]: Problem 8 (10th Edition)

Here's Q.8. below,

Consider a random sample X₁,X₂,...,X_n from a distribution with pdf f(x; θ) = θ(1 − x)^θ−1, 0 < x < 1, where 0 < θ. Find the form of the uniformly most powerful test of H₀: θ = 1 against H₁: θ > 1.

1. The owner of the Original Italian Pizzeria restaurant chain would like to predict the sales of her specialty deep-dish pizza. She has gathered data on the monthly sales of deep-dish pizzas at her restaurants and observations on other potentially relevant variables for each of her 15 restaurants in Northern California. The data is found in the file “DeepDishPizza.”

a. Create two (2) scatter charts. The first one will be a scatter chart of Quantity Sold (on the y-axis) and the variable Disposable Income per Household (on the x-axis). The second one will be a scatter chart of Quantity Sold (on the y-axis) and the variable Monthly Advertising Expenditures. Be sure to label the axes correctly.

b. Using the Regression function in Data Analysis, estimate a regression equation between Quantity Sold and Disposable Income per Household. Type your estimated equation in a text box. How would you explain the interpretation of the coefficient on Disposable Income per Household and the R 2 to the owner?

c. At a level of significance of 5%, test the null hypothesis that the coefficient on Disposable Income per Household is 0. Clearly explain your answer.

d. The owner also wants to know whether advertising expenses by her firm have any effect on the number of deep-dish pizzas sold. Using the Regression function in Data Analysis, estimate a regression equation between Quantity Sold and Disposable Income per Household Type and Monthly Advertising Expenditure. Type your estimated equation in a text box. How would you now explain both estimated coefficients and the R 2 to the owner?

e. At a level of significance of 5%, test the null hypothesis that the coefficient on each independent variable is 0. Clearly explain your answer.

f. For the equation you estimated in (d), what is the interpretation of the F-test? Is it significant at a level of significance of 5%?

g. Suppose the Disposable Income per Household is equal to $43,000. Using your estimated regression equation in (d), what should the owner plan on Monthly Advertising Expenditures per restaurant if she wants to sell 58,000 deep-dish pizza per restaurant? Show your work.

A researcher is curious if age makes a difference in whether or not students make use of the gym at a university. He takes a random sample of 30 days and counts the number of upperclassmen (Group 1) and underclassmen (Group 2) that use the gym each day. The data are below. The population standard deviation for underclassmen is known to be 22.57 and the population standard deviation for upperclassmen is known to be 13.57.

Upper Classmen average = 202.4, population SD = 13.57, n = 30

Under Classmen average = 191.3, population SD = 22.57, n = 30

Is there evidence to suggest that a difference exists in gym usage based on age? Construct a confidence interval for the data above to decide. Use α=0.10. Confidence Interval (round to 4 decimal places):

_____ < μ1 - μ2 < _____

Suppose a batch of steel rods produced at a steel plant have a mean length of 151 millimeters and a variance of 64.

If 116 rods are sampled at random from the batch, what is the probability that the mean length of the sample rods would be greater than 151.81 millimeters? Round your answer to four decimal places.

You wish to determine if there is a linear correlation between the two variables at a significance level of α=0.01α=0.01. You have the following bivariate data set.

What is the critival value for this hypothesis test?
r_c.v. =

What is the correlation coefficient for this data set?
r =

Your final conclusion is that...

• There is sufficient sample evidence to support the claim that there is a statistically significant correlation between the two variables.
• There is insufficient sample evidence to support the claim the there is a correlation between the two variables.

Note: Round to three decimal places when necessary.

A random sample of 75 pre-school children was taken. The child was asked to draw a nickel. The diameter of that nickel was recorded. Their parent's incomes (in thousands of $) and the diameter of the nickel they drew are given below.

Test the claim that there is significant correlation at the 0.01 significance level. Retain at least 3 decimals on all values.

a) Identify the correct alternative hypothesis.

• H1:ρ=0H1:ρ=0
• H1:r≠0H1:r≠0
• H1:μ≠0H1:μ≠0
• H1:pL≠pHH1:pL≠pH
• H1:ρ≠0H1:ρ≠0

b) The rr test statistic value is:   

c) The critical value is:

d) Based on this, we

• Reject H0H0
• Fail to reject H0H0

e) Which means

• There is sufficient evidence to warrant rejection of the claim
• There is not sufficient evidence to support the claim
• The sample data supports the claim
• There is not sufficient evidence to warrant rejection of the claim

f) The regression equation (in terms of income xx) is:
ˆy=y^=   

g) To predict what diameter a child would draw a nickel given family income, it would be most appropriate to:

• Use the regression equation
• Use the mean coin size
• Use the P-Value
• Use the residual

1. If n=10, ¯ x (x-bar)=43, and s=10, find the margin of error at a 95% confidence level (use at least two decimal places)

2. If n=11, ¯x (x-bar)=35, and s=4, find the margin of error at a 99% confidence level (use at least three decimal places)

A cell phone company offers a simple extended warranty plan. If your phone is damaged, they will repair it for up to $50. If you lose or destroy your phone, they will give you a $200 voucher towards a new phone. The company believes that 5% of customers will need the replacement voucher and 10% will request a repair. 1. If the company charges $25 for this extended warranty, what is the expected value of the profit they will earn? 2. What is the standard deviation of their profit? 3. Suppose the company collects 10 warranty plans on one day. What is the mean of the company's total profit? 4. What is the standard deviation of the 10 total warranty plans? What assumption does this calculation require? Do you think this assumption is reasonable? 5. What are the mean and standard deviation for the profit on a 1000 plans? 6. What do your answers to the previous question tell you about the company's likelihood of making a profit? 7. Is the $25 warranty a wise purchase for you? Given that you will probably buy dozens of devices over the next decade, are these types of warranties a wise purchase for you?

1. If n=28, x¯ (x-bar)=50, and s=6, find the margin of error at a 95% confidence level (use at least two decimal places)

2. What is the margin of error for a poll with a sample size of 2400 people? Round your answer to the nearest hundredth of a percent.

%

3. If you want a poll to have a margin of error of 3.24%, how large will your sample have to be? Round your answer to the nearest whole number.

__people