he International Air Transport Association surveys business travelers to develop quality ratings for transatlantic gateway airports. The maximum possible rating is 10. Suppose a simple random sample of 50 business travelers is selected and each traveler is asked to provide a rating for the Miami International Airport. The ratings obtained from the sample of 50 business travelers follow.

Develop a 95% confidence interval estimate of the population mean rating for Miami. Round your answers to two decimal places.

With the federalization of many offenses and increased Bureau of Prisons (BOP) responsibility for housing District of Columbia and immigration offenders, the BOP has grown every year from 2000 to 2013, when it peaked at 219,298. What factors influenced the continued significant growth of the BOP into the 2000s?

A company wants to determine whether its consumer product ratings

​(0minus−​10)

have changed from last year to this year. The table below shows the​ company's product ratings from eight consumers for last year and this year. At

alphaαequals=​0.05,

is there enough evidence to conclude that the ratings have​ changed? Assume the samples are random and​ dependent, and the population is normally distributed. Complete parts​ (a) through​ (f).

Which of the following situations describes a random variable that has a binomial distribution?

a)A fair coin is tossed 10 times. The variable X is the number of heads resulting from these 10 tosses.

b)A couple will keep having children until they have three girls or five children. The variable X is the number of children in the family.

c)The variable X is the number of clients in the bank between 10:00 a.m. and 11:00 a.m.

d)Fifteen cards are taken from the deck without placing them back in the deck. The variable X is the number of ace outcomes.

What is meant by “bias due to selective survival” in cross-sectional studies? (In your answer, make sure to define appropriate selection probability parameters.) Under what circumstances might there be no selective survival bias even if the selection probabilities are not all equal? Suppose that you could assess that the direction of possible selective survival bias in your study was towards the null. If your study data yielded a non-statistically significant odds ratio of 1.04, would it be correct to conclude that there was no exposure-disease association in your source population? Explain

2.Stats and Probability

1) 4 ballpoint pens are selected without replacement at random from a box that contains 2 blue pens, 3 red pens, and 5 green pens.
If X is the number of blue pens selected and Y is the number of red pens selected

a. Write the “joint probability distribution” of x and y.

b. Find P[(X, Y ) ∈ A], where A is the region
{(x, y)|x + y ≤ 2}.

c. Show that the column and row totals of a Table and give the marginal distribution of X alone and of Y alone

d. Find the conditional distribution of Y, given that X = 1; namely, f (Y|1)=?

e. Use it to determine P(Y = 0 | X = 1).

f. Show that if the random variables X and Y are statistically independent or NOT independent.

Please discuss among yourselves if we can prove or disprove a statistical hypothesis. For example, let's say that I have the following null hypothesis: "The population mean is equal to 5", and the alternative hypothesis: "The population mean is not equal to five". Then, I collect a sample from the population and after performing the hypothesis test I conclude at the 0.01 significance level: "Reject the null hypothesis", or in plain English "There is enough evidence to reject the claim that the population mean is equal to five". Then, am I proving that the population mean is not equal to five? Please comment about it and support your assertion (probably you have to do some research on Internet).

A study is conducted to compare salaries of managers of a certain industry employed

in two areas of the country, the eastern and northern regions. Independent ran

dom samples of 300 plant managers are selected for each of the two regions. These

managers were asked for their annual salaries. The results showed an average salary

x1 = $102300 and standard deviation s1 = $5700 for the eastern region and an average

salary x2 = $98500 and standard deviation s2 = $3800 for the northern region.

(a) Assuming normality for the distributions of annual salaries and the equality of

variances, construct a 99% confifidence interval for µ1 1 µ2, the difffference in the

mean salaries.

(b) Is the assumption of normality necessary? Why or why not?

(c) Is the assumption of equality of variances reasonable? Assume they are unequal

and obtain a 95% confifidence interval for the ratio of the two variances!

(d) If your answers to the previous questions are negative, compute a new 99%

confifidence interval for µ1 1 µ2 with the correct assumptions. Compare with the

result in (a).

(e) Let us assume that the data have not been collected yet. Let us also assume

that previous knowledge suggests that σ1 = σ2 = $4000. Are the sample sizes

of 300 suffiffifficient to produce a 95% confifidence interval on µ1 1 µ2 having a width

of only $ 1000?

2) X and Y have the following joint probability density function:

n=4, X=2, Y=1, Z=3

Find:

a)Marginal distribution of X and Y.

b)Mean of X and Y.

c)E(XY).

d)Covariance of X and Y and comment on it.

e)Correlation coefficient between X and Y. And comment.

2) X and Y have the following joint probability density function:

{((5y^3)/(96x^2)) 2<x<5, 0<y<4
       
    0      Elsewhere}

Find:
a) Marginal distribution of X and Y
b) Mean of X and Y
c) E(XY)
d) Covariance of X and Y and comment on it.
e) Correlation coefficient between X and Y. And comment.

A drug manufacturer uses two production facilities to produce a pain reliever. The amount of the active ingredient of the drug in the capsules at the two​ facilities, X₁ and X₂, are normally distributed random variables. The desire of the quality control manager is that the population mean amounts of the active ingredient in the​ capsules, μ₁ and μ₂​, be equal. Recent tests on small samples have indicated a noticeable increase in the amount of the active ingredient in capsules coming from Plant​ #1. The manager decides to select larger samples from each plant and test the hypotheses H₀: μ₁−μ₂ ≤ 0 and H_A​: μ₁−μ₂ ​> 0. The results from the 2 samples are given below. The manager is not willing to assume that the variances in the two groups are equal. Based on these​ results, which of the following is​ true?

X₁ = ​52.1; X₂ =​ 49.9; S₁ = ​2.3; S₂ = ​1.9; n₁ = ​40; n₂ = ​37; df = 74

A. If the null hypothesis is not rejected for α = ​.005, a Type II error has occurred.

B. For a level of significance of α = ​.01, the difference in the sample means is statistically significant.

C. Using a level of significance of α = ​.01, the null hypothesis should not be rejected.

D. For a level of significance of α = ​.005, a Type I error will be made if the null hypothesis is false.

E. The​ p-value for the test statistic is greater than .005.

A magazine collects data each year on the price of a hamburger in a certain fast food restaurant in various countries around the world. The price of this hamburger for a sample of restaurants in Europe in January resulted in the following hamburger prices (after conversion to U.S. dollars).

The mean price of this hamburger in the U.S. in January was $4.63. For purposes of this exercise, assume it is reasonable to regard the sample as representative of these European restaurants. Does the sample provide convincing evidence that the mean January price of this hamburger in Europe is greater than the reported U.S. price? Test the relevant hypotheses using α = 0.05. (Use a statistical computer package to calculate the P-value. Round your test statistic to two decimal places and your P-value to three decimal places.)

2. Distributions

a. For the standard normal probability distribution, what percent of the curve lies to the left of the mean?

b. Describe a normal distribution and a standard normal distribution.

c. You recently took a standardized test in which scores follow a normal distribution with a mean of 14 and a standard deviation of 5. You were told that your score is at the 75th percentile of this distribution. What is your score?

d. The random variable X is normally distributed with a mean of 14 and a standard deviation of 5. What is the probability that a value of X chosen at random will be between 7.5 and 18.5?

The wine quality dataset used in this chapter includes a variable on alcohol content. The average alcohol content of red wine is 10.423, while that of white wine is 10.5143. The respective standard deviations are 1.0657 and 1.2306. The respective sample sizes are 1599 and 4898.

2. What is the difference between means?

Research Scenario

Hypertension is defined as a systolic blood pressure (SBP) of 140 mmHg or more, or a diastolic blood pressure (DBP) of 90 mmHg or more. Diastolic blood pressure (DBP) measurements from the general population form a normal distribution with m = 82 mmHg and s = 10 mmHg. Epidemiological studies have shown that obesity increases the risk of hypertension (high DBP). Using data from a cohort study, a researcher obtained a sample of n = 144 obese persons whose average DBP was x̅ = 87. Do these data indicate that obesity is associated with hypertension (significantly higher DBP) in this sample? Test with alpha=.05.

Note

µ = population mean

x̅ = sample mean

s = standard deviation

---------------------------------------------------------------------------------------------------------------------

Note: Each response is worth 6 points

Step 1: Set up your hypothesis and determine the level of significance

            State the null hypothesis

1. In written format

Answer:

2. In mathematical format

Answer:

State the alternative hypothesis

3. In written format

Answer:

4. In mathematical format

Answer:

Determine the Level of significance

5. Based on information in the scenario, what is the level of significance to be used?

Answer:

Step 2: Select the appropriate test statistic

6. The appropriate test statistic for the above scenario is the one-sample z test. What is a Z test? Describe why this is the appropriate test for the scenario?

Answer:

Step 3: Set up the decision rule

7. Based on the level of significance you set in Step 1 (Question 5) and whether your alternative hypothesis is directional or non-directional, what is your decision rule? (In other words, what is your rejection region?)

Answer:

Step 4: Compute the test statistic

8. Calculate the z statistic. Please show your work.

Answer:

Step 5: Conclusion

9. Do you reject or fail to reject the null hypothesis?

Answer:

10. Is hypertension (high DBP) affected by obesity in this sample?

Answer:

Find the critical z-score value for each of the following confidence levels. Give each answer to at least three decimal places.

• The z-critical value for a 95% confidence level is:

  equation editor

  Equation Editor
• The z-critical value for an 80% confidence level is:

  equation editor

  Equation Editor
• The z-critical value for a 99% confidence level is:

  equation editor

  Equation Editor
• The z-critical value for a 92% confidence level is:

  equation editor

  Equation Editor
• The z-critical value for a 75% confidence level is:

  equation editor

  Equation Editor
• The z-critical value for an 85% confidence level is:

To construct a confidence interval for a population mean when the population standard deviation is unknown, find the critical t-score value for each of the following confidence levels, assuming a sample size of 12. Give each answer to at least three decimal places.

• The t-critical value for a 99% confidence level is:

  equation editor

  Equation Editor
• The t-critical value for a 95% confidence level is:

  equation editor

  Equation Editor
• The t-critical value for a 90% confidence level is:

  equation editor

  Equation Editor
• The t-critical value for a 98% confidence level is:

  equation editor

  Equation Editor
• The t-critical value for a 92% confidence level is: