State whether each of the following is always true (T) or not always true (F).

a) If X is a random variable, Corr X, (1/3)X= (1/3).

b) If X and Y are independent random variables then E(X|Y ) = E(X)

c) d) If fx(x) is the marginal density of a random variable X and fy(y|X = x) is the conditional density of a random variable Y , given a particular realization x of X, then the joint density of X and Y is given simply by f(x, y) = fx(x)fy(y|X = x)

d) If X1, X2, ..., Xn are independent random variables, each following a Bernoulli distribution with the same parameter p, then the sum Σn i=1Xi is a Binomial random variables, with parameters n and p.

e)If X1, X2, ..., X100 are independent, normally distributed random variables, then the average X¯ = 1 100Σ 100 i=1Xi of these random variables is itself a random variable following a normal distribution.

f) If Z1 and Z2 are independent random variables, each following a standard normal distribution, then Z1 + Z2 follows a standard normal distribution as well.

g) If X ∼ χ 2 (4) and Y ∼ t(1) then the 95th percentile of X exceeds the 95th percentile of Y .

h) If X ∼ F(5, 7) and Y ∼ F(7, 5), then the 5th percentile of X is greater than the 5th percentile of Y .

Refer to the CDI data set in Appendix C.2.
a. For each geographic region, regress the number of serious crimes in a CDI (Y) against
population density (X" total population divided by land area), per capita personal income
(X2 ), and percent high school graduates (X3 ). Use first-order regression model (6.5) with
three predictor variables. State the estimated regression functions.
b. Are the estimated regression functions similar for the four regions? Discuss.
c. Calculate MSE and R2 for each region. Are these measures similar for the four regions?
Discuss.
d. Obtain the residuals for each fitted model and prepare a box plot of the residuals for each
fitted modeL Interpret your plots and state your findings.

I Can do it by my self but Can you explain how what is meaning (b), (c) and (d)
Just write explantions

More than 100 million people around the world are not getting enough sleep; the average adult needs between 7.5 and 8 hours of sleep per night. College students are particularly at risk of not getting enough shut-eye.

A recent survey of several thousand college students indicated that the total hours of sleep time per night, denoted by the random variable X, can be approximated by a normalmodel with E(X) = 6.84 hours and SD(X) = 1.24 hours.

Question 1. Find the probability that the hours of sleep per night for a random sample of 4 college students has a mean x between 6.7 and 6.93.

(use 4 decimal places in your answer)

Question 2. Find the probability that the hours of sleep per night for a random sample of 16 college students has a mean x between 6.7 and 6.93.

(use 4 decimal places in your answer)

Question 3. Find the probability that the hours of sleep per night for a random sample of 25 college students has a mean x between 6.7 and 6.93.

(use 4 decimal places in your answer)

Question 4. The Central Limit Theorem was needed to answer questions 1, 2, and 3 above.

TrueFalse    

1A) Twenty laboratory mice were randomly divided into two groups of 10. Each group was fed according to a prescribed diet. At the end of 3 weeks, the weight gained by each animal was recorded. Do the data in the following table justify the conclusion that the mean weight gained on diet B was greater than the mean weight gained on diet A, at the α = 0.05 level of significance? Assume normality. (Use Diet B - Diet A.)

(a) Find t. (Give your answer correct to two decimal places.)


(b) Find the p-value. (Give your answer correct to four decimal places.)

1B) A bakery is considering buying one of two gas ovens. The bakery requires that the temperature remain constant during a baking operation. A study was conducted to measure the variance in temperature of the ovens during the baking process. The variance in temperature before the thermostat restarted the flame for the Monarch oven was 2.2 for 23 measurements. The variance for the Kraft oven was 3.2 for 21 measurements. Does this information provide sufficient reason to conclude that there is a difference in the variances for the two ovens? Assume measurements are normally distributed and use a 0.02 level of significance.

(a) Find F. (Give your answer correct to two decimal places.)


(b) Find the p-value. (Give your answer correct to four decimal places.)

Given the following sample information, test the hypothesis that the treatment means are equal at the 0.10 significance level:

Complete the ANOVA table. (Round the SS, MS, and F values to 2 decimal places.)

Please explain how you got your answer. I am very new to statistics so the clearer the better.

How do you evaluate each of the assumptions of regression analysis?

A) The proportion of subjects who answered the last question of their survey as "Very Satisfied" is 0.80. The hospital is hoping to increase that proportion to 0.9 and they consider a change in the proportion of 0.10 to be significant. For an alpha of 0.05 and a power of 80%, calculate the number of surveys they would need to collect in order to detect a difference. 4) Proportion of subjects who answered the last question of their survey as "Very Satisfied" is 0.80. The hospital is hoping to increase that proportion to 0.9 and they consider a change in the proportion of 0.10 to be significant. For an alpha of 0.01 and a power of 80%, calculate the number of surveys they would need to collect in order to detect a difference.

B) Proportion of subjects who answered the last question of their survey as "Very Satisfied" is 0.80. The hospital is hoping to increase that proportion to 0.9 and they consider a change in the proportion of 0.10 to be significant. For an alpha of 0.01 and a power of 80%, calculate the number of surveys they would need to collect in order to detect a difference.

The Excel file BankData shows the values of the following variables for randomly selected 93 employees of a large bank. (A very similar 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 with the 99% confidence level (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 monthly 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.

In order to investigate if the systolic blood pressure measurements vary in standing and lying positions the systolic blood pressure levels of a sample of 12 persons were measured in both positions (first in standing position and then in lying position). The data of this experiment are recorded in SSPS file.

1. A researcher believes the mean systolic blood pressure level in standing position is more than that in lying position. Test the appropriate hypothesis to investigate the researcher’s belief at alpha=0.05.

2. Find the 95% confidence interval for the difference in mean systolic blood pressure levels in two positions. Interpret the result.

Please mentioned the steps in SPSS and screen shoot of it

The table is for the data.

From 104 of its restaurants, Noodles & Company managers collected data on per-person sales and the percent of sales due to "potstickers" (a popular food item). Both numerical variables failed tests for normality, so they tried a chi-square test. Each variable was converted into ordinal categories (low, medium, high) using cutoff points that produced roughly equal group sizes. At α = .10, is per-person spending independent of percent of sales from potstickers? Potsticker % of Sales Per-Person Spending Low Medium High Row Total Low 14 13 8 35 Medium 11 17 5 33 High 10 8 18 36 Col Total 35 38 31 104 You will need to open the Excel file. Then open Minitab. Copy the data (NOT THE TOTALS) into Minitab. Be sure that the 1st number goes into row 1 in Minitab and that you type the column headings (Low, Medium, High) into the grey shaded top header row in Minitab. PictureClick here for the Excel Data File (a) The hypothesis for the given issue is H0: Percentage of Sales and Per-Person Spending are independent. No Yes (b) Calculate the chi-square test statistic, degrees of freedom, and the p-value. (Round your test statistic value to 2 decimal places and p-value to 4 decimal places. Leave no cells blank - be certain to enter "0" wherever required.) Test statistic d.f. p-value (c) We reject the null and find dependence. No Yes

The following data lists the ages of a random selection of actresses when they won an award in the category of Best​ Actress, along with the ages of actors when they won in the category of Best Actor. The ages are matched according to the year that the awards were presented. Complete parts​ (a) and​ (b) below.

a. Use the sample data with a

0.010.01

significance level to test the claim that for the population of ages of Best Actresses and Best​ Actors, the differences have a mean less than 0​ (indicating that the Best Actresses are generally younger than Best​Actors).In this​ example,

mu Subscript dμd

is the mean value of the differences d for the population of all pairs of​ data, where each individual difference d is defined as the​ actress's age minus the​ actor's age. What are the null and alternative hypotheses for the hypothesis​ test?

Upper H 0H0​:

mu Subscript dμd

equals=

00 ​year(s)

Upper H 1H1​:

mu Subscript dμd

less than<

00 ​year(s)

​(Type integers or decimals. Do not​ round.)

Identify the test statistic.

tequals=negative 3.68−3.68

​(Round to two decimal places as​ needed.)

Identify the​ P-value.

​P-valueequals=0.0020.002

​(Round to three decimal places as​ needed.)

What is the conclusion based on the hypothesis​ test?

Since the​ P-value is

less than or equal to

the significance​ level,

reject

the null hypothesis. There

is

sufficient evidence to support the claim that actresses are generally younger when they won the award than actors.

b. Construct the confidence interval that could be used for the hypothesis test described in part​ (a). What feature of the confidence interval leads to the same conclusion reached in part​ (a)?

The confidence interval is

nothing

​year(s)less than<mu Subscript dμdless than<nothing

​year(s).

​(Round to one decimal place as​ needed.)

What feature of the confidence interval leads to the same conclusion reached in part​ (a)?

Since the confidence interval contains

zero,

only positive numbers,

only negative numbers,

reject

fail to reject

the null hypothesis.

Actress (years)   Actor (years)
30   58
30   41
31   35
28   42
33   31
27   34
25   46
42   38
29   41
31   43

Question 1:Professor Handy measured the time in seconds required to catch a falling meter stick for 12 randomly selected students’ dominant hand and nondominant hand. The Minitab Express file contains these measurements. Professor Handy claims that the reaction time in an individual’s dominant hand is less than the reaction time in their nondominant hand. Assuming that the differences follow a normal distribution, test the claim at the 5% significance level.

Question 2:The New England Patriots. The 2017 roster of the New England Patriots, winners of the 2017 NFL Super Bowl included 12 defensive linemen and 9 offensive linemen. The Minitab Express file for this problem contains the weights in pounds of the offensive and defensive linemen. Use this data set to test the claim that the defensive linesmen weigh less that the offensive linemen at the 5% level of significance.

Question 3: 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.)

Please provide all answers in the following format:

Step 1: State the null and alternative hypothesis. (Use “mu” for the symbol μ.)

Step 2: Calculate the test statistic.

Step 3: Find the p-value.

Step 4: State your conclusion. (Do not just say “Reject H0” or “Do not reject H0.” State the conclusion in the context of the problem.)