Suppose that the score in a qualifying exam for a certain profession is normally distributed with mean 80 and standard deviation 36. Use Excel to find the following probabilities:

b) Find the probability that a student gets a score between 50 and 100.

c) To get an A, a student must score in the top 8%. What is the minimum score needed to get an A?

d) Find the probability that the average score in a sample of 60 students is between 50 and 100?

e) Graph the distributions of the random variable in c) and d) and explain why the answers in b) and d) are different.

Please answer using excel functions.

The Excel file BankData shows the values of the following variables for randomly selected 93 employees of a large bank. This real data set was used in a court lawsuit against discrimination.  

Let

= monthly salary in dollars (SALARY),

= years of schooling at the time of hire (EDUCAT),

= number of months of previous work experience (EXPER),

= number of months that the individual was hired by the bank (MONTHS),

= dummy variable coded 1 for males and 0 for females (MALE).

Using the t-test studied in Section 10.2, you could find some evidence that the mean salary of all male employees is greater than the mean salaries of all female employees, and hence provide some support for a discrimination suit against the employer. It is recognized, however, that a simplecomparison of the mean salaries might be insufficient to conclude that the female employees have been discriminated against. Obviously there are other factors that affect the salary. These factors have been identified as  and  defined above.

Assume the following multiple linear regression model,

,

and apply Regression in Data Analysis of Excel (see pages 312 – 314) to find the estimated regression equation

.

Note. Of course, Input Y Range is A1:A94, Input X range is B1:E94, and Labels should be checked.

1. Clearly show the estimated regression equation. What is the percentage of variation in the salary explained by this equation? Assuming that the values of  and  are fixed, what is the estimated difference between the predicted salaries of male and female employees?

2. What salary would you predict for a male employee with 12 years educations, 10 months of previous work experience, and with the time hired equal to 15 months? What salary would you predict for a female employee with 12 years educations, 10 months of previous work experience, and with the time hired equal to 15 months? What is the difference between the two predicted salaries? Compare this difference with that stated in Task 1.

3. Is there a significant difference in the predicted salaries for male and female employees after accounting for the effects of the three other independent variables? To answer this question, conduct the ttest for the significance of at a 1% level of significance. Clearly show the null and alternative hypotheses to be tested, the value of the test statistic, the p-value of the test, your conclusion and its interpretation; see pages 322 – 323 and 333 – 335.

Suppose the proportion of left-handed individuals in a population is θ. Based on a simple random sample of 20, you observe four left-handed individuals. Using R

a) Assuming the sample size is small relative to the population size, plot the loglikelihood function and determine the Maximum Likelihood Estimate.

b) If instead the population size is only 50, then plot the log-likelihood function and determine the MLE. (Hint: Remember that the number of left-handed individuals follows a hypergeometric distribution. This forces θ to be of the form i/50 for some integer i between 4 and 34. From a tabulation of the log-likelihood, you can obtain the MLE.)

1. Stress and weight in rats. In a study of the effects of stress on weight in rats, 71 rats were randomly assigned to either a stressful environment or a control (nonstressful) environment. After 21 days, the change in weight (in grams) was determined for each rat. The table below summarizes the data on weight gain. Test the claim that stress effects weight. (Use a 10% significance level.)

Paired-/Related Samples T-test

Use it to test whether there is a difference between the two conditions(note: conditions must be RELATED-participants provide in each condition).

You are interested in the relationship satisfaction of young adults before and after they go off to college/university, which separates them from their sweetheart. You asked four couples to rate the satisfaction of their relationship (on a scale of 0-50) before leaving for school and then again after a semester. Here are the data:

Pair Before After

1 40 32

2 38 31

3 36 30

4 42 31

1. State the null and alternative hypotheses as well as your criterion:

2 .State your assumptions

3. Calculate difference scores, the sum of difference scores and the sum of difference scores squared:

4. Calculate t

5. Figure out your degrees of freedom and use this to find the critical t value

6. Reject or fail-to-reject the null hypothesis and state your conclusions.

There are three urns each containing seven red, five green, and three white balls, and two old urns each containing five red, three green, and seven white balls. The urns are identical except for an old or new date stamped beneath the base. If a single red ball is randomly drawn from one of these urns, was it most probably drawn from an old urn or a new urn?

Dr. Xiong, a clinical psychologist, wishes to test the claim that there is a significant difference in a person's adult weight if he is raised by his father instead of his mother. Dr. Xiong surveys five sets of identical twin boys who were raised separately, one by the mother and one twin by the father. Each twin is weighed and identified as having been raised by his mother or his father. The following table lists the results. Do these data support Dr. Xiong's claim at the 0.01 level of significance?

Weights of Twins (in Pounds)

Step 1 of 3:

State the hypotheses for this test.
Answers: A:  H0: μ₁ ≤ μ_{2 ,} Ha: μ₁ > μ₂
B: H0: μ_d = 0, Ha: μ_d ≠ 0
C: H0: μ_d ≤ 5, Ha: μ_d > 5
D: H0: μ₁ = μ_2, Ha: μ₁ ≠ μ₂

Step 2 of 3
Compute the value of the test statistic.
Answers: A:  F = 2.12
B: t = 9.213
C: F = 3.695
D: t = -.370

Step 3 of 3: State the conclusion for this test.
Answers: A: Since p-value < 0.10, reject H0. There is sufficient evidence to support the claim that there is a significant difference in a person's weight if he is raised by his father rather than his mother.
B: Since t < -4.604, reject H0. There is sufficient evidence to support the claim that there is a significant difference in a person's weight if he is raised by his father rather than his mother.
C: Since p-value > 0.10, fail to reject H0. There is not sufficient evidence to support the claim that there is a significant difference in a person's weight if he is raised by his father rather than his mother.
D: Since t > -4.604, fail to reject H0. There is sufficient evidence to support the claim that weight and who raised the child are related.

In the binomial probability distribution, let the number of trials be n = 4, and let the probability of success be p = 0.3310. Use a calculator to compute the following.

(a) The probability of three successes. (Round your answer to three decimal places.)


(b) The probability of four successes. (Round your answer to three decimal places.)


(c) The probability of three or four successes. (Round your answer to three decimal places.)

An inspector has the job of checking a screw-making machine at the start of each day. She finds that the machine needs repairs pairs one day out of ten. When the machine does need repairs, all the screws it makes are defective. Even when the machine is working properly, 5% of the screws it makes are defective; these defective screws are randomly scattered through the day's output. Use a calculator to get approximate answers to the following lowing questions. What is the probability that the machine is in good order if:

a. The first screw the inspector tests is defective?

b. The first two screws are both defective?

c. The first three are all defective?

A. A new shopping mall is considering setting up an information desk manned by one employee. Based on information obtained from similar information desks, it is believed that people will arrive at the desk at the rate of 20 per hour. It takes an average of 2 minutes to answer a question. It is assumed that arrivals are Poisson and answer times are exponentially distributed. i. Find the probability that the employee is idle. ii. Find the proportion of the time that the employee is busy. iii. Find the average number of people receiving and waiting to receive information. iv. Find the average number of people waiting in line to get information. v. Find the average time a person seeking information spends at the desk. vi. Find the expected time a person spends just waiting in line to have a question answered

The numbers of words defined on randomly selected pages from a dictionary are shown below. Find the​ mean, median, and mode of the listed numbers. Or is there no mean, median or mode?

38  53  67  54  34  58  55  43  31  39

If n = 19, ¯ x = 43, and s = 17, construct a confidence interval at a 80% confidence level. Assume the data came from a normally distributed population. Give your answers to three decimal places.

< μ

Use the following information to answer the next questions: We are interested in whether this age group of males fits the distribution of the U.S. adult population. Calculate the frequency one would expect when surveying 400 people.
Observed Values: Never married 150; Married 228; Widowed 5; Divorced/Separated 17
Expected Values follow these proportions: Never married 30%; Married 56%; Widowed 3%; Divorced/Separated 11%

What test are you running?

What is the observed values for never married?

What is the observed values for married?

What is the observed values for widowed?

What is the observed values for divorced/separated?

What is the expected values for never married?

What is the expected values for married?

What is the expected values for widowed?

What is the expected values for divorced/separated?

What are the degrees of freedom?

What is the null hypothesis?

What is the alternative hypothesis?

What is the test statistic? Use one decimal place.

What is the p-value? Use three decimal places.

What is your conclusion based on the p-value and the level of significance?

At the 5% significance level, what can you conclude?

The target value for a watch is to lose no time over a year. A sample of 4 watches had the following time gained (lost) in minutes: (-1, -2, 3, -3). Assume k = $4. Calculate the following:

Round your answers to two decimal places.

Average squared deviation:

Average loss per watch:

Total expected loss for 8,000 watches produced and sold: $