Sarah is the owner of Sarah’s Pub, and her best business is on Friday and Saturday nights when customers buy plenty of alcoholic beverages (alcohol). After studying a large sample of receipts from Friday and Saturday nights, Sarah knows the following. 73% of her orders on these nights involve at least one alcohol sale. 23% of her customers order only alcohol, 28% have dinner plates, and the rest have sandwiches.   Of those who order dinner plates, 60% order alcohol; and of those who order sandwiches, 89% order alcohol. Sarah still has questions, as do I.

1. (5 points) What is the probability that a randomly selected customer orders a sandwich?

Type the answer and any work below. Write one sentence interpreting the result.

2. (5 points) What is the probability a randomly selected customer ordered alcohol and a

sandwich? Type the answer and any work below. Write one sentence

interpreting the result.

3. (5 points) What is the probability a randomly selected customer ordered alcohol or a

sandwich? Type the answer and any work below. Write one sentence

interpreting the result.

4. (5 points) What is the probability a randomly selected orders alcohol and a dinner plate? Type

the answer and any work below. Write one sentence interpreting the result.

5. (5 points) Are orders for alcohol and orders for dinner plates independent events? Type the

answer and any work below. You must report two probabilities and explain your

reasoning to receive credit for this answer.

A survey of 250 students is selected randomly on a large university campus. They are asked if they use a laptop in class to take notes. The result of the survey is that 115 of the 250 students responded​ "yes." An approximate 95​% confidence interval is (0.397​,0.523​). Which of the following are​ true? If they are not​ true, briefly explain why not

a)95​% of the students fall in the interval ​(0.397​,0.523​).

​b) The true proportion of students who use laptops to take notes is captured in the interval left parenthesis 0.397 comma 0.523 right parenthesis with probability 0.95.

​c) There is a 46​% chance that a student uses a laptop to take notes.

​d) There is a 95​% chance that the student uses a laptop to take notes 46​% of the time.

​e) We are 95​% confident that the true proportion of students who use laptops to take notes is captured in the interval left parenthesis 0.397 comma 0.523 right parenthesis.

a) Choose the correct answer below.

A. The statement is false. 46​% of the students fall in the interval ​(0.397​,0.523​).

B. The statement is false. This​ doesn't make sense because students are not proportions.

C. The statement is true.

D. The statement is false. The sample size is too small to make this claim.

​b) Choose the correct answer below.

A. The statement is false. The population proportion is either within the interval or not within the interval.

B. The statement is true.

C. The statement is false. The sample size is too small to make this claim.

D. The statement is false. We are 46​% confident that the sample proportion of students who use a laptop to take notes is between 39.7​% and 52.3​%. ​

c) Choose the correct answer below.

A. The statement is false. There is not enough information to make an absolute statement about the population value with precision.

B. The statement is true.

C. The statement is false. There is a​ 5% chance that a random selected student uses a laptop to take notes.

D. The statement is false. The sample size is too small to make this claim.

​d) Choose the correct answer below.

A. The statement is false. There is a 54​% chance that a random selected coworker uses a laptop to take notes.

B. The statement is true.

C. The statement is false. The sample size is too small to make this claim.

D. The statement is false. This​ doesn't make sense because​ it's not about the proportion of the time that a student uses a laptop. ​

e) Choose the correct answer below.

A. The statement is false. We are 46​% confident that between 39.7​% and 52.3​% of the samples will have a proportion near 95​%.

B. The statement is false. The sample size is too small to make this claim.

C. The statement is false. The statement should be about the true​ proportion, not future samples.

D. The statement is true.

Each day in April, you have an independent 1/4 chance of deciding to take a 6am run.

1. (a) What is the probability you go on exactly 12 runs in the month of April (which has

   30 days)?

2. (b) What is the expected number of days you go running during April?

3. (c) What is the probability that you go running at least once during April 1–7?

4. (d) What is the probability that that your first run of the month occurs on April 5?

5. (e) What is the probability that your first run of the month occurs on or before April 20?

Jobs and productivity! How do retail stores rate? One way to answer this question is to examine annual profits per employee. The following data give annual profits per employee (in units of 1 thousand dollars per employee) for companies in retail sales. Assume σ ≈ 3.7 thousand dollars.

(a) Use a calculator or appropriate computer software to find x for the preceding data. (Round your answer to two decimal places.)
thousand dollars per employee

(b) Let us say that the preceding data are representative of the entire sector of retail sales companies. Find an 80% confidence interval for μ, the average annual profit per employee for retail sales. (Round your answers to two decimal places.)

(c) Let us say that you are the manager of a retail store with a large number of employees. Suppose the annual profits are less than 3 thousand dollars per employee. Do you think this might be low compared with other retail stores? Explain by referring to the confidence interval you computed in part (b).

Yes. This confidence interval suggests that the profits per employee are less than those of other retail stores.No. This confidence interval suggests that the profits per employee do not differ from those of other retail stores.    

(d) Suppose the annual profits are more than 6.5 thousand dollars per employee. As store manager, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (b).

Yes. This confidence interval suggests that the profits per employee are greater than those of other retail stores.No. This confidence interval suggests that the profits per employee do not differ from those of other retail stores.    

(e) Find an 95% confidence interval for μ, the average annual profit per employee for retail sales. (Round your answers to two decimal places.)

Let us say that you are the manager of a retail store with a large number of employees. Suppose the annual profits are less than 3 thousand dollars per employee. Do you think this might be low compared with other retail stores? Explain by referring to the confidence interval you computed in part (b).

Yes. This confidence interval suggests that the profits per employee are less than those of other retail stores.No. This confidence interval suggests that the profits per employee do not differ from those of other retail stores.    

Suppose the annual profits are more than 6.5 thousand dollars per employee. As store manager, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (b).

Yes. This confidence interval suggests that the profits per employee are greater than those of other retail stores.No. This confidence interval suggests that the profits per employee do not differ from those of other retail stores.    

Masterfoods USA states that their color blends were selected by conducting consumer preference tests, which indicated the assortment of colors that pleased the greatest number of people and created the most attractive overall effect. On average, they claim the following percentages of colors for M&Ms® milk chocolate candies: 24% blue, 20% orange, 16% green, 14% yellow, 13% red and 13% brown.

5. Test their claim that the true proportion of red M&Ms® candies is 0.13 at the 0.05 significance level.

6. Test their claim that the true proportion of brown M&Ms® candies is 0.13 at the 0.05 significance level.

7. On average, they claim that a 1.69 oz bag will contain more than 54 candies. Test this claim (µ > 54) at the 0.01 significance (σ unknown).

HELP:

As an example, say we had found 732 purple candies out of 3500 total candies. The sample proportion of purple candies is 732/3500 = 0.2091428571.
Now let's say you want to test that the true proportion of purple candies is 21% (0.21).
First define your hypotheses:
H0: p = 0.21 (claim)
H1: p ≠ 0.21

Next we need to calculate the test statistic. For this type of test, it is a z and a two tailed test. You have been asked to test at alpha = 0.05, so we will reject the null if the test statistic, z, is positive and greater than 1.96 OR if the test statistic, z, is negative and smaller than -1.96. (NOTE: This is the same as if the absolute value of the test statistic is greater than 1.96.)

Review: → sample proportion (0.209143)
p → assumed value in null (0.21)
q → 1 - p (0.79)
n → total number of candies (3500)

Because the test statistic is negatiae and is NOT smaller than -1.96, we FAIL TO REJECT. We have insufficient evidence to suggest the true proportion is not 0.21.

You will follow this procedure for EACH color.

Use a one-way ANOVA test to study the price difference between cruelty-free products and those tested on animals

• Select at least seven different products for your one-way ANOVA test: for example, you may select laundry detergent or mascara. Share your results.
• Are there significant price differences?
• Explain whether or not there are other factors to consider in the price differences between your selected products.

Suppose an environmental agency would like to investigate the relationship between the engine size of​ sedans, x, and the miles per gallon​ (MPG), y, they get. The accompanying table shows the engine size in cubic liters and rated miles per gallon for a selection of sedans. The regression line for the data is y hat=36.7920−4.1547x.

Use this information to complete the parts below.

Engine Size   MPG
2.4 27
2.1 31
2.3 26
3.4 22
3.5 24
2.2 28
2.2 24
2.1 29
3.9 20

a) Calculate the coefficient of determination. R²=? (Round to three decimal places as​ needed.)

b) Using α=0.05​, test the significance of the population coefficient of determination.

Determine the null and alternative hypotheses.

c) The​ F-test statistic is? ​(Round to two decimal places as​needed.)

d) the p-value is? ​(Round to three decimal places as​needed.)

e) Construct a​ 95% confidence interval for the average MPG of a 2.5​-cubic liter engine.

UCL= ? ​(Round to two decimal places as​ needed.)

LCL= ​? (Round to two decimal places as​ needed.)

f) Construct a​ 95% prediction interval for the MPG of a 2.5​-cubic liter engine.

UPL= ? ​(Round to two decimal places as​ needed.)

LPL= ? ​(Round to two decimal places as​ needed.)

Suppose that the random variables, ξ, η have joint uniform density f(x, y) = 2/9

in the triangular region bounded by the lines x = -1 , y - -1 and y = 1 - x.

a) Find the marginal densities f(x) =∫ 2/9 dy (limits, -1 to 1-x) and f(y) =∫ 2/9 dx

(limits -1 to 1-y). Also show that f(x) f(y) ≠ f(x, y) so that ξ and η are not

independent.

b) Verify that μ_ξ = ∫ x f(x) dx = 0 and μ_η = ∫ y f(y) dy = 0

c) Find Var (ξ) = ∫ x² f(x) dx, Var (ζ) = ∫ y² f(y) dy and

Cov (ξ, η) = 2/9 ∫ x [ ∫ y dy ] dx (y limits -1 to 1-x, then x = -1 to x = 2)

d) Find ρ and regression curve E[η│ξ = x] = [1/f(x)] (2/9) ∫ y dy (y= -1 to y = 1-x)

For what type of dependent variable is logistic regression appropriate? Give an example of such a variable. In what metric are logistic regression coefficients? What can we do to them to make them more interpretable, and how would we interpret the resulting translated coefficients?

(Understanding and Using Statistics for Criminology and Criminal Justice)

Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car. They randomly survey 410 drivers and find that 283 claim to always buckle up. Construct a 91% confidence interval for the population proportion that claim to always buckle up.

The lifetimes of a certain electronic component are known to be normally distributed with a mean of 1,400 hours and a standard deviation of 600 hours. For a random sample of 25 components the probability is 0.6915 that the sample mean lifetime is less than how many hours?

A)1345

B)1460

C)1804

D)1790

In testing the hypothesis H₀:  μ = 800 vs. H_a:  μ ≠ 800. A sample of size 40 is chosen. the sample mean was found to be 812.5 with a standard deviation of 25, then the of the test statistic is:

In a sample of 500 voters, 400 indicated they favor the incumbent governor.  The 95% confidence

interval of voters not favoring the incumbent is

A Type II error is committed if we make:

What is the value of z _α/2 for an 85% confidence interval?

A random sample of size 15 taken from a normally distributed population revealed a sample mean of 75 and a sample variance of 25. The upper limit of a 95% confidence interval for the population mean would equal:

A genetic experiment with peas resulted in one sample of offspring that consisted of 417 green peas and 153 yellow peas.

a. Construct a 90% confidence interval to estimate of the percentage of yellow peas.

b. It was expected that​ 25% of the offspring peas would be yellow. Given that the percentage of offspring yellow peas is not​ 25%, do the results contradict​ expectations?

The Damon family owns a large grape vineyard in western New York along Lake Erie. The grapevines must be sprayed at the beginning of the growing season to protect against various insects and diseases. Two new insecticides have just been marketed: Pernod 5 and Action. To test their effectiveness, three long rows were selected and sprayed with Pernod 5, and three others were sprayed with Action. When the grapes ripened, 440 of the vines treated with Pernod 5 were checked for infestation. Likewise, a sample of 360 vines sprayed with Action were checked. The results are:

At the 0.10 significance level, can we conclude that there is a difference in the proportion of vines infested using Pernod 5 as opposed to Action? Hint: For the calculations, assume the Pernod 5 as the first sample.

a) state the decision rule

b) compute the pooled proportion

c) Compute the value of the test statistic

d) What is your decision regarding the null hypothesis?