Let X₁, X₂,..., X_nbe independent and identically distributed exponential random variables with parameter λ .

a) Compute P{max(X₁, X₂,..., X_n) ≤ x} and find the pdf of Y = max(X₁, X₂,..., X_n).
b) Compute P{min(X₁, X₂,..., X_n) ≤ x} and find the pdf of Z = min(X₁, X₂,..., X_n).
c) Compute E(Y) and E(Z).

A Pew Research Center survey asked respondents if they would rather live in a place with a slower pace of life or a place with a faster pace of life. The survey also asked the respondent’s gender. Consider the following sample data.

a. Is the preferred pace of life independent of gender? Using a  level of significance, what is the -value?

Compute the value of the  test statistic (to 3 decimals).

The -value is - Select your answer -between .01 and .025between .025 and .05between .05 and .10greater than .10less than .01Item 2

What is your conclusion?

- Select your answer -Cannot concludeConcludeItem 3 that the preferred pace of life is not independent of gender.

b. What are the percentage responses for each gender (to 1 decimal)?

Discuss any differences between the preferences of men and women.

- Select your answer -Men onlyWomen onlyBoth men and womenItem 10 prefer a slower pace of life. Women have a higher preference for a  - Select your answer -slowerfasterItem 11 pace of life, while men have a higher preference for a - Select your answer -slowerfasterItem 12 pace of life.

A survey found that​ women's heights are normally distributed with mean 62.4 in. and standard deviation 2.1 in. The survey also found that​ men's heights are normally distributed with mean 67.3 in. and standard deviation 3.1 in. Most of the live characters employed at an amusement park have height requirements of a minimum of 56 in. and a maximum of 63 in.

1.Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as characters at the amusement​ park?

2.Find the percentage of women meeting the height requirement. What does the result suggest about the genders of the people who are employed as characters at the amusement​ park?

Logistic regression predicts a 1._____________, 2._____________, 3.______________from one or more categorical or continuous predictor variables.

When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ.

Method 1: Use the Student's t distribution with d.f. = n − 1.
This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method.

Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the standard normal distribution.
This method is based on the fact that for large samples, s is a fairly good approximation for σ. Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution.

Consider a random sample of size n = 41, with sample mean x = 46.0 and sample standard deviation s = 4.6.

(a) Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

(b) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

(c) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

Yes. The respective intervals based on the t distribution are shorter.

Yes. The respective intervals based on the t distribution are longer.    

No. The respective intervals based on the t distribution are shorter.

No. The respective intervals based on the t distribution are longer.

(d) Now consider a sample size of 81. Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

(e) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

(f) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

No. The respective intervals based on the t distribution are longer.

Yes. The respective intervals based on the t distribution are shorter.   

No. The respective intervals based on the t distribution are shorter.

Yes. The respective intervals based on the t distribution are longer.

With increased sample size, do the two methods give respective confidence intervals that are more similar?

As the sample size increases, the difference between the two methods remains constant.

As the sample size increases, the difference between the two methods becomes greater.    

As the sample size increases, the difference between the two methods is less pronounced.

Compare the mean and standard deviation for the Coin variable (question 2) with those of the mean and standard deviation for the binomial distribution that was calculated by hand in question 5. Explain how they are related in a short paragraph of several complete sentences.

Mean from question #2: 4.543

Standard deviation from question #2: 1.521

Mean from question #5: 5

Standard deviation from question #5: 1.581

2.The weights of four randomly and independently selected bags of potatoes labeled 20 pounds were found to be 21, 22​, ,20.4, and 21.2

Assume Normality.

a. Using a​ two-sided alternative​ hypothesis, should you be able to reject the hypothesis that the population mean is 20

pounds using a significance level of 0.05

Why or why​ not? The confidence interval is reported​ here: I am

95​%confident the population mean is between 20.1 and 22.2pounds.

b. Test the hypothesis that the population mean is not 20

Use a significance level of 0.05

c. Choose one of the following​ conclusions:

i. We cannot reject a population mean of 20 pounds.

ii. We can reject a population mean of 20 pounds.

iii. The population mean is 21.15 pounds.

A pizza restaurant monitors the size (measured by the diameter) of the 10-inch pizzas that it prepares. Pizza crusts are made from doughsthat are prepared and prepackaged in boxes of 15 by a supplier. Doughsare thawed and pressed in a pressing machine. The toppings are added, and the pizzas are baked. The wetness of the doughsvaries from box to box, and if the dough is too wet or greasy, it is difficult to press, resulting in a crust that is too small. The first shift of workers begins work at 4 P.M., and a new shift takes over at 9 P.M. and works until closing. The pressing machine is readjusted at the beginning of each shift. The restaurant takes five consecutive pizzas prepared at the beginning of each hour from opening to closing on a particular day. The diameter of each baked pizza in the subgroups is measured, and the pizza crust diameters obtained are given in Table.

Use the pizza crust diameter data to do the following:

a. Show that X bar =10.028 and R bar = 0.84.

b. Find the center lines and control limits for the X bar and Rcharts for the pizza crust data.

c. Set up the X bar and Rcharts for the pizza crust data.

d. Is the R chart for the pizza crust data in statistical control? Explain.

e. Is the X bar chart for the pizza crust data in statistical control? If not, use the X bar chart and the information given with the data to try to identify any assignable causes that might exist.

f. Suppose that, based on the X bar chart, the manager of the restaurant decides that the employees do not know how to properly adjust the dough pressing machine. Because of this, the manager thoroughly trains the employees in the use of this equipment. Because an assignable cause (incorrect adjustment of the pressing machine) has been found and eliminated, we can remove the subgroups affected by this unusual process variation from the data set. We therefore drop subgroups 1 and 6 from the data. Use the remaining eight subgroups to show that we obtain revised center lines of X bar = 10.2225 and R bar = 0.825.

g. Use the revised values of and to compute revised and R chart control limits for the pizza crust diameter data. Set up X bar and R charts using these revised limits. Be sure to omit subgroup means and ranges for subgroups 1 and 6 when setting up these charts.

h. Has removing the assignable cause brought the process into statistical control? Explain.

Dana uses the following parameters to determine that she needs a sample size of 140 participants for her study that will compare the means of two independent groups (t-test) using a one-tailed hypothesis:

            Effect size (d) = .5

            alpha = .05

            Power (1 - beta) = .90

Using the above information, answer each of the following questions.

a. If Dana keeps alpha and sample size the same, but desires an effect size of .80, what will happen to power? Will it increase or decrease? Explain.

b. If Dana keeps the desired effect size and sample size the same, but reduces alpha to .025, what will happen to power? Will it increase or decrease? Explain.

c. If Dana keeps power and sample size the same, but increases the desired effect size to .8, what will happen to alpha? Will it increase or decrease? Explain.

d. Dana decides she wants to increase the desired effect size to 2.0 and increase the power to .99, but keeps alpha the same. She does so to increase the likelihood that an effect will be found, and to make sure her results demonstrate a large enough effect. Also, when she conducts her a priori power analysis to determine her sample size, she is excited to see that she needs far fewer participants in her sample with those parameters (target n = 18). What, if any, are the problems with Dana’s strategy?

Answer the questions as indicated for scenarios 1-5. Note: You do not need to perform any of the procedures indicated.

Scenario 3: A guidance counselor identifies a random sample of 40 high school female students and gives each of these students a vocabulary test. For the female group, the average vocabulary score was 69 with a standard deviation of 5.3. Next, the guidance counselor takes a random sample of 48 male high school students. The male students also complete the vocabulary test. This group had an average vocabulary score of 64 with a standard deviation of 5.6. What is a 90% confidence interval estimate for difference in average vocabulary test score between female and male students at this school?

A. Indicate whether the inference procedure needed is a confidence interval or a significance test.

B. Indicate whether the procedure involves one or two samples.

C. Name the inference method needed to answer the question posed.

D. Verify whether or not the conditions have been met for this inference procedure. Specifically, you need to list each condition and then explain how the condition was or was not met.

E. Determine if it is appropriate to perform the significance test or confidence interval (yes/no).

Find the mean, mod, median, and standard deviation of the following data. And Based on these results, check whether the value of 10 is usual? 5, 6, 7, 8, 9,8,7,8 _________________________________________________________________________ Pre-Employment Drug Screening Results are shown in the following Table: Positive Test Result Negative Test Result Subject Uses Drugs 8 (True Positive) 2 (False Negative) Subject is not a Drug User 10 (False Positive) 180 (True Negative) If 1 of the 200 test subjects is randomly selected, find the probability that the subject had a positive test result, given that the subject actually uses drugs. That is, find (positive test result subject uses drugs). If 1 of the 200 test subjects is randomly selected, find the probability that the subject actually uses drugs, given that he or she had a positive test result. That is, find ( subject uses drugs positive test result ). _______________________________________________________________________ This is observation from previous years about the impact of students working while they are enrolled in classes, due to students too much work, they are spending less time on their classes. First, the observer need to find out, on average, how many hours a week students are working. They know from previous studies that the standard deviation of this variable is about 5 hours. A survey of 200 students provides a sample mean of 7.10 hours worked. What is a 95% confidence interval based on this sample?

A survey was conducted among 20 adults. The following shows the age of the respondents.

44 47 47 47 47 52 53 53 54 54

55 56 57 58 58 64 66 66 69 83

(1) Please calculate the mean, median, 1st quartile (i.e. 25th percentile), 3rd quartile (i.e. 75th percentile), and IQR for their age. (2 points for each question, 10 points in total)

(2) For the above 20 adults, are there outliers (i.e. are there people with extreme age)? Please show your calculations for identifying the outliers. (2 points)

2.Thirty GPAs from a randomly selected sample of statistics students at a college are linked below. Assume that the population distribution is approximately Normal. The technician in charge of records claimed that the population mean GPA for the whole college is 2.81. a. What is the sample​ mean? Is it higher or lower than the population mean of 2.81​? b. The chair of the mathematics department claims that statistics students typically have higher GPAs than the typical college student. Use the​ four-step procedure and the data provided to test this claim. Use a significance level of 0.05.

2.86,3.37,3.17,2.51,3.49, 2.75, 3.04, 3.59,2.65,3.97,2.89,2.66,3.52,3.06, 2.79,3.45,2.47,3.14,3.43,3.13,3.18,3.08,3.09,2.96,3.49, 3.43,2.73,3.14, 3.11,3.03

In Country​ A, the population mean height for​ 3-year-old boys is 37 inches. Suppose a random sample of 15​ 3-year-old boys from Country B showed a sample mean of 36.5 inches with a standard deviation of 4 inches. The boys were independently sampled. Assume that heights are Normally distributed in the population. Complete parts a through c below. a. Determine whether the population mean for Country B boys is significantly different from the Country A mean. Use a significance level of 0.05. Find test statistic and p-value b. Now suppose the sample consists of 30 boys instead of 15 and repeat the test. Find the test statistic and p- value