Construct  90%, 95%, and 99% confidence intervals for the population mean, and state the practical and probabilistic interpretations of each. Indicate which interpretation you think would be more appropriate to use. Explain why the three intervals that you construct are not of equal width. Indicate which of the three intervals you would prefer to use as an estimate of the population mean, and state the reason for your choice.

In a length of hospitalization study conducted by several cooperating hospitals, a random sample of 64 peptic ulcer patients was drawn from a list of all peptic ulcer patients ever admitted to the participating hospitals and the length of hospitalization per admission was determined for each. The mean length of hospitalization was found to be 8.25 days. The population standard deviation is known to be 3 days.

For each confidence interval indicate

a) Z-critical values ,

b) Error,

c) confidence interval

1. Keno: Keno game is a game with 80 numbers 1, 2, … , 80 where 20 numbered balls out of these 80 numbers will be picked randomly. You can pick 4, 5, 6, or 12 numbers as shown in the attached Keno payoff / odds card.

When you pick 4 numbers, there is this 4-spot special that you place $2.00 in the bet, and you are paid $410 if all your 4 numbers are among the 20 numbers, or your are paid $4.00 if 3 of the 4 numbers are among the 20 numbers chosen from the 80 numbers.

The number of ways of picking 20 numbers from 80 is C(80, 20) = 80! / (60! 20!).

The number of ways that all your 4 numbers are among the 20 numbers is: C(76, 16) (why?) = 76! / (60! 16!).

The probability that your 4 numbers bingo is C(76, 16) / C(80, 20), and the theoretical payoff should be C(80, 20) / C(76, 16) = 80! * 16! / (76! * 20!) = (80 * 79 * 78 * 77 ) / (20 * 19 * 18 * 17 ) = $326.4355…

1. (6%) Based on this computation, is the payoff fair? Explain!
2. (6%) The payoff for 3 numbers in your 4 chosen numbers appear in the 20 numbers is $4.00. Is that a fair payoff (how is the number of ways of 3 numbers matching related to the number of ways of 4 numbers matching?)?

In a certain congressional district, it is known that 40% of the registered voters classify themselves as conservatives. If ten registered voters are selected at random from this district, what is the probability that two of them will be conservatives? (Round your answer to four decimal places.)

What is the best way to show and identify the Ho and H1 of mental illness with explaining what “significance” is in a general sense and in the chosen article. discuss the p-value.

Scenario: Imagine that a psychologist working with veterans with post-traumatic stress disorder wants to compare the effectiveness of several therapies focused on reducing symptoms of anxiety. The psychologist randomly sampled 20 veterans who recently returned from combat and randomly assigned each of them to receive one of four interventions for 8 weeks. A survey was used to measure the participants’ anxiety at the end of the 8 weeks. Higher anxiety scores indicate more anxiety. You can find the data for this Assignment in the Weekly Data Set forum found on the course navigation menu.

1. Identify the obtained F value using SPSS and report it in your answer document.
2. Identify the p value using SPSS and report it in your answer document.
3. Explain whether the F test is statistically significant. Explain how you know.
4. Explain what the psychologist can conclude about the relationship between different types of therapy and anxiety.
5. Should the psychologist conduct a post hoc test? Why or why not? If post hoc testing is needed, conduct a Tukey HSD post hoc analysis in SPSS. Explain what the results tell you.

Crusty’s has collected data for the last ten days for the time from when an order is taken until the pizza is finished (see follow data) (five samples are taken each day).

1. Which of the following is true regarding the range?
   1. The upper control limit is 35.02
   2. The lower control limit is 21.47
   3. The lower control limit is 5.64
   4. The upper control limit is 16.56

Iron-deficiency anemia is the most common form of malnutrition in developing countries, affecting about 50% of children and women and 25% of men. Iron pots for cooking foods had traditionally been used in many of these countries, but they have been largely replaced by aluminum pots, which are cheaper and lighter. Some research has suggested that food cooked in iron pots will contain more iron than food cooked in other types of pots. One study designed to investigate this issue compared the iron content of some Ethiopian foods cooked in aluminum, clay, and iron pots. Foods considered were yesiga wet', beef cut into small pieces and prepared with several Ethiopian spices; shiro wet', a legume-based mixture of chickpea flour and Ethiopian spiced pepper; and ye-atkilt allych'a, a lightly spiced vegetable casserole. Four samples of each food were cooked in each type of pot. The iron in the food is measured in milligrams of iron per 100 grams of cooked food. The data are shown in the table below.

(a) Make a table giving the sample size, mean, and standard deviation for each type of pot. Is it reasonable to pool the variances? Although the standard deviations vary more than we would like, this is partially due to the small sample sizes, and we will proceed with the analysis of variance.

(b) Plot the means. Give a short summary of how the iron content of foods depends upon the cooking pot.

(c) Run the analysis of variance. Give the ANOVA table, the F statistics with degrees of freedom and P-values, and your conclusions regarding the hypotheses about main effects and interactions.

I used this code for parts a-d:

x=runif(1e6,1,5)
mean(x)
var(x)
hist(x,main="Histogram of x",xlab="x",ylab="f(x)",border="blue",col="green",freq=F)

Write R code that does the following: Let Xi be a random variable uniformly distributed between 1 and 5. (a) Find E[Xi ] and Var(Xi).

(b) Generate 1000000 samples from a random variable X that has a uniform density on [1,5]. x=runif(1e6,1,5)

(c) Create a histogram of the distribution. hist(x,freq=F) Explain the shape of histogram. Is it consistent with the uniformly distributied rv between 1 and 5? See how to add details such as colors and labels in Homeworks 3, 5.

d) Now we will withdraw n samples out of distribution and take the average. We will repeat the process 1e6 times! and make a histogram of the sampling distribution.

I used this R code for part e through f:Sample_Avg <- function(t){mean(runif(t,1,5))}
avg(30) > 3

rep(avg(30),1e6)

e) In a script file create a function Sample_Avg that will withdraw n samples and take the average. Sample Avg <- function(t){ mean(runif(t,1,5) } Run Sample Ave(30) This will be the average of 30 samples

f) Use replicate command to create 1000000 of these sample averages and save results in Xavg. Be patient it will take about a min.

g) Create a histogram of Xavg. Wow! it looks different compared to the original distribution. Why is that? What is the shape of this sampling distribution? What are the parameters.

h) Overlay the distribution function from g) onto the histogram of Xave. See Homeworks 3,5 for commands for overlaying function on top of a histogram.

i) Compute the theoretical probability P(X₃₀ > 3.5) .In R how would you find the percentage of Xavg that is greater than 3.5?? Execute the code and compare it to the theoretical calculation.

Please answer g, h and i. Thank you

Suppose it is known that in a certain population the mean systolic blood pressure (SBP) is 120 mmHg and the standard deviation is 10 mmHg. In a random sample of size 40 from this population, what is the probability that this sample will have a mean SBP greater than 124 mmHg?

A certain railway company claims that its trains run late 5 minutes on the average. The actual times (minutes) that 10 randomly selected trains ran late were provided giving a sample mean = 9.130 and sample standard deviation s = 1.4 . In testing the company’s claim, (2-sided test) at the significance level of 0.01 and assuming normality Find a 99% confidence interval and state which of the following is true about trains belonging to this railway company?

A. We conclude that the trains run late an average of 9.13 minutes.

B. The correct answer is not among the choices.

C. We would reject the claim that the trains run late an average of 5 minutes and conclude that they run late an average of more than 5 minutes since the 99% confidence interval exceeds 5.

D. We do not have enough evidence to reject the claim that the trains run late an average of 5 minutes since the value of 5 is included in the 99% confidence interval.

E. We would reject the claim that the trains run late an average of 5 minutes and conclude that they run late an average of less than 5 minutes since 5 exceeds the entire 99% confidence interval.

100 losses are independent, identically distributed, with a common uniform distribution over (0, 6). Using the normal approximation, calculate the probability that the sum of all the losses (the aggregate loss) will exceed 275.

Please use R to do it.

Using the SATGPA data set in Stat2Data package. Test by using α= .05

Question: Test if the proportion of MathSAT greater than VerbalSAT is 0.60

> library(Stat2Data)
> data("SATGPA")
> data(SATGPA)
> SATGPA

As a criminologist, you are interested in the factors related to juvenile offending. You think that time spent studying influences delinquent behavior. In a group of 21 high school students, you collect information about the number of times a juvenile has broken curfew this month and how much time they spend studying this month. You break the sample into three groups, those who spend no time studying, those who spend less than 1 hour per day, and those who spend more than 1 hour per day).

a) What is the independent variable?              

b) What is the dependent variable?

c) What is the null hypothesis?

d) What is the research hypothesis?

e) Interpret the results.

Test whether the proportion of iphone owners is more than the proportion of android owners. Take two samples of at least 20 each. In the first sample ask "Do you own an iphone?" in the second sample ask "do you own an android?" Use a significance level of 0.01.

a. survey results: iphone -> 13 yes, 7 no android -> 11 yes, 9 no

b. state your claim

c. null hypothesis and alternative hypothesis

d. which type of test are you running? show calculator input.

e. P-value

f. Decision: reject null or fail to reject null

g. conclusion

h. calculate a 98% confidence interval to estimate the difference in proportions of iphone and android users. write it in sentence format.

i. does the confidence interval agree or contradict your hypothesis test conclusion? why?