Below we are investigating the racial differences in occupational prestige. You will be asked to determine the area under the standard normal curve and present proportions, percentages, and counts based on that information. HINT: You’re being asked to compute z-scores and investigate their corresponding areas here. Have APPENDIX B ready!

The table below contains information on the occupational prestige scores for black and white Americans:

Mean Standard deviation N

Black Americans 40.83 13.07 195

White Americans 45.03 13.93 1,100

Use sentences to respond to the following. Please Number each of your responses, show your work for all computations, and underline your final answers. Calculate and report your answers with two decimal points unless your answers come directly from APPENDIX B.

What proportion of white Americans have an occupational prestige score above (greater than) 60? How would you write this as a percentage? Approximately how many people is this in the sample? (7.5 Points)

What proportion of black Americans have an occupational prestige score below (less than) 60? How would you write this as a percentage? Approximately how many people is this in the sample? (7.5 Points)

What proportion of white Americans occupational have an occupational prestige score between 30 and 70? How would you write this as a percentage? Approximately how many people is this in the sample? (7.5 Points)

What proportion of black Americans occupational have an occupational prestige score between 30 and 70? How would you write this as a percentage? Approximately how many people is this in the sample? (7.5 Points)

To see for yourself how the central limit theorem works, let's say we have a normal distribution (with mean =100 and standard devation = 20). Let's generate some random samples of various sizes from this distribution. We can do this in excel using =norm.inv(rand(),100,20) and it will randomly generate numbers from this distribution. I generated four samples of size 5, 10, 20 and 30, and got the means of 124 (n=5); 91 (n=10); 105 (n=20); 103 (n=30). If I continue to increase the sample size, my average values should converge to the mean of 100. Now you try. Pick a distribution and generate some sample sizes to prove this to yourself. Post and discuss your results.

An assistant in the district sales office of a national cosmetics firm obtained data on advertising expenditures and sales last year in the district’s 44 territories.

X1: expenditures for point-of-sale displays in beauty salons and department stores (X$1000).

X2: expenditures for local media advertising.

X3: expenditures for prorated share of national media advertising.

Y: Sales (X$1000).

1. Test the regression relation between sales and the three predictor variables. State the hypotheses, test statistic and degrees of freedom, the p-value, the conclusion in words.

2. Determine whether the linear regression model is appropriate by using the “usual” plots (scatterplot, residual plots, histogram/QQ plot). Explain in detail whether or not each assumption appears to be substantially violated.

For each of the examples below, draw (or describe drawing) a sampling distribution around the reported mean, mark the upper and lower limits of the 95% confidence interval, and compute the mean values that correspond to those upper and lower limits.

1. Dr. Grimm collected a random sample of 30 sociology majors and administered an IQ test (σ = 15). This group had an average IQ of M = 105. Compute a 95% confidence interval to determine the population mean of psychology majors’ IQ scores.

2. According to 2016 data from the Centers for Disease Control and Prevention, the average life expectancy in the United States is 78.6 years, with an approximate standard deviation, σ, of 15 years. A random sample of 45 dead rock musicians finds an average age of death of 45.2 years. Calculate a 95% confidence interval to determine the population mean of a rock musician’s life expectancy.

3. Dr. Morris collects information on text messaging from a random sample of 50 adults ages 25 to 44. Dr. Morris finds that these individuals send or receive an average of 75 text messages per day. Using a standard deviation, σ, of 22, calculate the 95% confidence interval for the average number of texts sent by adults in this age group.

A random sample of ? measurements was selected from a population with standard deviation ?=16.6 and unknown mean ?. Calculate a 90 % confidence interval for ? for each of the following situations: (a) ?=65, ?⎯⎯⎯=86.1 (b) ?=80, ?⎯⎯⎯=86.1 (c) ?=100, ?⎯⎯⎯=86.1

Two plots at Rothamsted Experimental Station were studied for production of wheat straw. For a random sample of years, the annual wheat straw production (in pounds) from one plot was as follows.

Use a calculator to verify that, for this plot, the sample variance is s² ≈ 0.272.

Another random sample of years for a second plot gave the following annual wheat production (in pounds).

Use a calculator to verify that the sample variance for this plot is s² ≈ 1.052.

Test the claim that there is a difference (either way) in the population variance of wheat straw production for these two plots. Use a 5% level of signifcance.

(a) What is the level of significance?

State the null and alternate hypotheses.

H_o: σ₁² = σ₂²; H₁: σ₁² > σ₂²

H_o: σ₁² > σ₂²; H₁: σ₁² = σ₂²    

H_o: σ₂² = σ₁²; H₁: σ₂² > σ₁²

H_o: σ₁² = σ₂²; H₁: σ₁² ≠ σ₂²

(b) Find the value of the sample F statistic. (Use 2 decimal places.)


What are the degrees of freedom?

What assumptions are you making about the original distribution?

The populations follow independent normal distributions.

The populations follow independent normal distributions. We have random samples from each population.    

The populations follow independent chi-square distributions. We have random samples from each population.

The populations follow dependent normal distributions. We have random samples from each population.

(c) Find or estimate the P-value of the sample test statistic. (Use 4 decimal places.)

p-value > 0.200

0.100 < p-value < 0.200    

0.050 < p-value < 0.100

0.020 < p-value < 0.050

0.002 < p-value < 0.020

p-value < 0.002

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.    

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is sufficient evidence that the variance in annual wheat production differs between the two plots.

Reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production differs between the two plots.    

Reject the null hypothesis, there is sufficient evidence that the variance in annual wheat production differs between the two plots.

Fail to reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production differs between the two plots.

1.) A recent study indicated that 85% of all parents take candy from their child's trick-or-treat bag. If 30 parents are selected at random then what is the probability that 25 of them took Halloween candy from their child's trick-or-treat bag? Give your answer to four decimal places.

2.) Systolic blood pressure readings for females are normally distributed with a mean of 125 and a standard deviation of 10.34. If 60 females are randomly selected then find the probability that their mean systolic blood pressure is between 122 and 126. Give your answer to four decimal places.

3.) Apartment rental prices in Pittsburgh are approximately normally distributed with a mean of $838 and a standard deviation of $175. If a researcher wants to study people whose rent is in the lower 5% then find the maximum rent a person can pay and be part of the study. Give your answer to two decimal places and do not give units.

An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities.

If required, round your answers to two decimal places.

In a new promotional activity at Starbucks, whenever you buy a coffee, they give you fortune cookie, on which there is the digit 1,2, or 3, i.e., one of them. So whenever, you buy a coffee, you look at the number and if it is a number that you do not have it, then you collect that number. When you collect the numbers 1,2, and 3, then Starbucks gives you a free coffee.

(a) If X represents the number of coffees you need to purchase in order to get a free coffee, determine PMF of X for X values only until 5. No need to get the PMF for the other values of X since there will be two many cases.

(b) As we can see in Part a, the PMF calculation will be too hectic for the next values of X. Hence, the calculation for the parameters like mean and variance would be difficult. However, we can determine the mean of this distribution by using the concept of geometric distribution. After modeling your problem at hand with geometric distribution, determine how many coffees you need to purchase to get a free coffee?

Question 4

It has been hypothesized that the distribution of seasonal colds in Canada is as follows:

A random sample of 200 Canadian citizens provided the following results:

10 marks     Do the observed data contradict the hypothesis? Formulate and test the appropriate hypotheses at the 5% level of significance. Use the critical value approach.

1. Indicate which type of t-test would be appropriate. You can choose between the one sample t-test, dependent samples t-test and independent samples t-test. Write down the null and alternative hypotheses.

(a) In a sample of 20 newborn Russian Blue kittens, the mean weight was 3 ounces with a standard deviation of 0.5 ounces. In a sample of 20 newborn Turkish Van kittens, the mean weight was 5 ounces with a standard deviation of 0.6 ounces. On average, are the birth weights of Russian Blue kittens different from those into Turkish Van kittens?

(b) 12 normal (“wild type”) and 13 cyclin d2 knockout mice (mutants) were placed in devices which recorded their locomotor activity. The mean and standard deviation of activity for the former group were 112 and 36, while those for the latter were 200 and 25. Do the two mouse genotypes differ in average amount of locomotor activity?

(c) A sample of 25 students from a certain school have a mean SAT score of 1200 points. Suppose the general population of test-takers has a mean of 1060 points and a standard deviation of 110 points. On average, do students from the school in question score higher than the general population?

(d) Mice used in research come in different strains (a bit like breeds of dog or cat). These can differ in characteristics such as the startle response, i.e. how forcefully the animal flinches when it hears a loud sudden noise. Some researchers wanted to find a mouse strain with a strong startle response as a basis for a line of mutants. They therefore compared the startle response in 10 mice of the C57BL/6J strain and 10 mice of the 129X1/SvJ strain.

(e) Amphetamine has been found in some cases to reduce the startle response of mice. Quinpirole is drug that acts on some of the same neurotransmitters as amphetamine but is more selective and has a different mechanism. Researchers wondered therefore whether quinpirole would reduce the startle response in mice. They therefore injected 10 mice with quinpirole and 10 mice with saline (as a control) and measured their startle response to loud noises.

(f) Imagine the same situation as (e) above, except that instead of injecting each mouse only once with either quinpirole or saline, all 20 mice are given both injections on separate days. For example, mouse #1 gets saline and has his startle response tested, then several days later receives quinpirole and his startle response is tested again.

(g) Now imagine the same situation as (e) above except that all 20 mice are tested once, with quinpirole, and their startle response is com- pared to the previously published population mean and standard deviation of startle responses for mice without quinpirole (this is not how you would do the experiment in real life).

For large U.S. companies, what percentage of their total income comes from foreign sales? A random sample of technology companies (IBM, Hewlett-Packard, Intel, and others) gave the following information.†

Another independent random sample of basic consumer product companies (Goodyear, Sarah Lee, H.J. Heinz, Toys 'R' Us) gave the following information.

Assume that the distributions of percentage foreign revenue are mound-shaped and symmetric for these two company types.

(a) Use a calculator with mean and standard deviation keys to calculate x₁, s₁, x₂, and s₂. (Round your answers to two decimal places.)

(b) Let μ₁ be the population mean for x₁ and let μ₂ be the population mean for x₂. Find a 98% confidence interval for μ₁ − μ₂. (Round your answers to two decimal places.)

An experimental surgical procedure is being studied as an alternative to the old method. Both methods are considered safe. Five surgeons preform the operation on two patients matched by age, sex, and other relevant factors, with the results shown. The time to complete the surgery (in minutes) is recorded. At the 5% significance level, is the new way faster?

*Do not use p-values

*Use and show the 5 step method

D-bar=

S_d=

Step 1: H₀=

H_A=

Step 2: alpha =

Step 3: Test Statistic:

Step 4: Decision Rule:

Step 5: Calculation and Decision

Reject or do not reject H₀? Why?

The population average cholesterol content of a certain brand of egg is 215 milligrams, and the standard deviation is 15 milligrams. Assume the variable is normally distributed.

If we are told the average for 25 eggs is less than 220 mg, what is the probability that the average is less than 210?

Round the answers to four decimal places.