Complete all of the steps to derive the normal equations for simple linear regression and then solve them.

A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time exceeds 7.7 hours.

A. 0.8531

B. 0.9634

C. 0.9712

D. 0.9931

A sample of 11 individuals shows the following monthly incomes.

PLEASE DON’T COPY PASTE FEOM PREVIOUS QUESTION

1)On Planet Geometry, whenever two right angles have children they can have rectangles or squares with equal probability each. Consider a (very nice) pair of right angles that have 2 children.

a) Using a tree diagram, what is the probability that both children are squares given that at least one is a square? (It is not 1⁄2!)

2) Suppose that P(A∩B)=.3,P(A)=.6, and P(B)=.5.

a. Are A and B mutually exclusive?
b. Are A and B independent?

A recent headline announced / impeachment causes a dip in a President’s Approval Rating.
This conclusion was based on a University poll of 100 adults. In the poll, 41.1% of respondents approved of the president’s job performance.

(a) Based on this poll, what is the probability that more than 50% of the population approve of the president’s job performance?
(b) Form a 90% confidence interval for this estimate.

(c) Assume we know without sampling error the population support was 42% before the impeachment story broke in September. Test whether the impeachment proceedings have actually caused a dip in approval from the 42% baseline at the .05 significance level.

Problem 1 (3 + 3 + 3 = 9) Suppose you draw two cards from a deck of 52 cards without replacement. 1) What’s the probability that both of the cards are hearts? 2) What’s the probability that exactly one of the cards are hearts? 3) What’s the probability that none of the cards are hearts?

Problem 2 (4) A factory produces 100 unit of a certain product and 5 of them are defective. If 3 units are picked at random then what is the probability that none of them are defective?

Problem 3 (3+4=7) There are 3 bags each containing 100 marbles. Bag 1 has 75 red and 25 blue marbles. Bag 2 has 60 red and 40 blue marbles. Bag 3 has 45 red and 55 blue marbles. Now a bag is chosen at random and a marble is also picked at random. 1) What is the probability that the marble is blue? 2) What happens when the first bag is chosen with probability 0.5 and other bags with equal probability each?

Probem 4 (3+3+4=10) Before each class, I either drink a cup of coffee, a cup of tea, or a cup of water. The probability of coffee is 0.7, the probability of tea is 0.2, and the probability of water is 0.1. If I drink coffee, the probability that the lecture ends early is 0.3. If I drink tea, the probability that the lecture ends early is 0.2. If I drink water, the lecture never ends early. 1) What’s the probability that I drink tea and finish the lecture early? 2) What’s the probability that I finish the lecture early? 3) Given the lecture finishes early, what’s the probability I drank coffee?

Problem 5 (4+4+4=12) We roll two fair 6-sided dice. Each one of the 36 possible outcomes is assumed to be equally likely. 1) Find the probability that doubles were rolled. 2) Given that the roll resulted in a sum of 4 or less, find the conditional probability that doubles were rolled. 3) Given that the two dice land on different numbers, find the conditional probability that at least one die is a 1. Problem 6 (8) For any events A, B, and C, prove the following equality: P(B|A) P(C|A) = P(B|A ∩ C) P(C|A ∩ B)

A European growth mutual fund specializes in stocks from the British Isles, continental Europe, and Scandinavia. The fund has over 375 stocks. Let x be a random variable that represents the monthly percentage return for this fund. Suppose x has mean μ = 1.2% and standard deviation σ = 0.5%.

(a) Let's consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all European stocks. Is it reasonable to assume that x (the average monthly return on the 375 stocks in the fund) has a distribution that is approximately normal? Explain. , x is a mean of a sample of n = 375 stocks. By the , the x distribution approximately normal.

(b) After 9 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? (Round your answer to four decimal places.)

(c) After 18 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? (Round your answer to four decimal places.)

(d) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased? Why would this happen?

Yes, probability increases as the mean increases.

Yes, probability increases as the standard deviation decreases.

No, the probability stayed the same.

Yes, probability increases as the standard deviation increases.

(e) If after 18 months the average monthly percentage return x is more than 2%, would that tend to shake your confidence in the statement that μ = 1.2%? If this happened, do you think the European stock market might be heating up? (Round your answer to four decimal places.)

P(x > 2%) = ?????

Explain. This is very likely if μ = 1.2%. One would suspect that the European stock market may be heating up.

This is very likely if μ = 1.2%. One would not suspect that the European stock market may be heating up.

This is very unlikely if μ = 1.2%. One would not suspect that the European stock market may be heating up.

This is very unlikely if μ = 1.2%. One would suspect that the European stock market may be heating up.

You are interested in determining whether an experimental birth control pull has the side effects of changing blood pressure. You plan to randomly sample 18 women from the city in which you live. You give half of them a placebo for a month and then measure their diastolic blood pressure. Then you switch them to the birth control pill for a month and again measure their blood pressure. The other 9 women receive the same treatment except they are given the birth control pill first for a month, followed by the placebo for a month. The blood pressure readings are shown here. What is the power of your study to reject the null assuming that the alternative hypothesis is true. Assume alpha =.05 non directional alternative hypothesis. YOU DO NOT NEED ANY DATA IT ONLY LOOKS FOR POWER

What is the power to detect a false null hypothesis assuming Preal=.9. Put your work and answer below.

What is the probability of failing to reject the null if the null is false assuming Preal =.9. Put your work and answer below.

What is the power to detect a false null hypothesis assuming Preal=.7. Put your work and answer below.

What is the probability of making a type 2 error assuming Preal =.7. Put your work and answer below.

The following data represent petal lengths (in cm) for independent random samples of two species of Iris. Petal length (in cm) of Iris virginica: x1; n1 =

35 5.1 5.8 6.5 6.1 5.1 5.5 5.3 5.5 6.9 5.0 4.9 6.0 4.8 6.1 5.6 5.1 5.6 4.8 5.4 5.1 5.1 5.9 5.2 5.7 5.4 4.5 6.4 5.3 5.5 6.7 5.7 4.9 4.8 5.9 5.1

Petal length (in cm) of Iris setosa: x2; n2 =

38 1.6 1.8 1.4 1.5 1.5 1.6 1.4 1.1 1.2 1.4 1.7 1.0 1.7 1.9 1.6 1.4 1.5 1.4 1.2 1.3 1.5 1.3 1.6 1.9 1.4 1.6 1.5 1.4 1.6 1.2 1.9 1.5 1.6 1.4 1.3 1.7 1.5 1.6

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

x1 =

s1 =

x2 =

s2 =

(b) Let μ1 be the population mean for x1 and let μ2 be the population mean for x2. Find a 99% confidence interval for μ1 − μ2. (Round your answers to two decimal places.)

lower limit

upper limit

(c) Explain what the confidence interval means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the 99% level of confidence, is the population mean petal length of Iris virginica longer than that of Iris setosa?

Because the interval contains only positive numbers, we can say that the mean petal length of Iris virginica is longer.

Because the interval contains only negative numbers, we can say that the mean petal length of Iris virginica is shorter.   

Because the interval contains both positive and negative numbers, we cannot say that the mean petal length of Iris virginica is longer.

(d) Which distribution did you use? Why?

The Student's t-distribution was used because σ₁ and σ₂ are unknown.

The standard normal distribution was used because σ₁ and σ₂ are unknown.

The Student's t-distribution was used because σ₁ and σ₂ are known.

The standard normal distribution was used because σ₁ and σ₂ are known.

Do you need information about the petal length distributions? Explain.

Both samples are large, so information about the distributions is not needed

.Both samples are large, so information about the distributions is needed.

Both samples are small, so information about the distributions is needed.

Both samples are small, so information about the distributions is not needed.

​If, based on a sample size of 850​, a political candidate finds that 471 people would vote for him in a​ two-person race.

a. A 90% confidence interval for his expected proportion of the vote is ____ , ____

b. Would he be confident of winning based on this​ poll?

A. How well do people remember their past diet? Data are available for 91 people who were asked about their diet when they were 18 years old. Researchers asked them at about age 55 to describe their eating habits at age 18. For each subject, the researchers calculated the correlation between actual intakes of many foods at age 18 and the intakes the subjects now remember. The median of the 91 correlations was r = 0.217. The researchers stated, "We conclude that memory of food intake in the distant past is fair to poor".

Choose the best reason why r = 0.217 points to this conclusion.

Because a correlation of 0.217 indicates a negative association.

Because a correlation of 0.217 indicates a positive association.

Because a correlation of 0.217 indicates a strong association.

Because a correlation of 0.217 indicates a weak association.

Because a correlation of 0.217 indicates no association.

B. Although research questions usually concern a _________, the actual research is typically conducted with a ________.

sample, statistic

population, parameter

sample, population

population, sample

What is the model for this linear programing problem?

USA Today reported that approximately 25% of all state prison inmates released on parole become repeat offenders while on parole. Suppose the parole board is examining five prisoners up for parole. Let x = number of prisoners out of five on parole who become repeat offenders. x 0 1 2 3 4 5 P(x) 0.217 0.368 0.220 0.156 0.038 0.001 (a) Find the probability that one or more of the five parolees will be repeat offenders. (Round your answer to three decimal places.) Incorrect: Your answer is incorrect. How does this number relate to the probability that none of the parolees will be repeat offenders? These probabilities are the same. This is twice the probability of no repeat offenders. This is five times the probability of no repeat offenders. This is the complement of the probability of no repeat offenders. These probabilities are not related to each other. (b) Find the probability that two or more of the five parolees will be repeat offenders. (Round your answer to three decimal places.) (c) Find the probability that four or more of the five parolees will be repeat offenders. (Round your answer to three decimal places.) (d) Compute μ, the expected number of repeat offenders out of five. (Round your answer to three decimal places.) μ = prisoners (e) Compute σ, the standard deviation of the number of repeat offenders out of five. (Round your answer to two decimal places.) σ = prisoners

The manufacturers of Good-O use two different types of machines to fill their 25 kg packs of dried dog food. On the basis of random samples of size 15 and 18 from output from machines 1 and 2 respectively, the mean and standard deviation of the weight of the packs of dog food produced were found to be 28.99 kg and 0.142 kg for machine 1 and 26.376 kg and 0.383 kg for machine 2. Hence, under the usual assumptions, determine a 95% confidence interval for the difference between the average weight of the output of machine 1 and machine 2. Use machine 1 minus machine 2, stating the upper limit of the interval correct to three decimal places.