The housing market has recovered slowly from the economic crisis of 2008.​ Recently, in one large​ community, realtors randomly sampled 29 bids from potential buyers to estimate the average loss in home value. The sample showed the average loss was ​$9440 with a standard deviation of ​$2030. In​ 2011, the average home in this region of the country lost ​$8874 in value. Was the community studied by the realtors​ unusual? Use a​ t-test to decide if the average loss observed was significantly different from the regional average with 0.05 as the​ P-value cutoff level.

An article in Journal of the American Statistical Association (1990, Vol. 85, pp. 972–985) measured weight of 30 rats under experiment controls. Suppose that there are 12 underweight rats. (a) Calculate a 90% two-sided confidence interval on the true proportion of rats that would show underweight from the experiment. Round your answers to 3 decimal places. Enter your answer; confidence interval, lower bound ≤p≤ Enter your answer; confidence interval, upper bound (b) Using the point estimate of p obtained from the preliminary sample, what sample size is needed to be 90% confident that the error in estimating the true value of p is no more than 0.02? n= Enter your answer in accordance to the item b) of the question statement (c) How large must the sample be if we wish to be at least 90% confident that the error in estimating p is less than 0.02, regardless of the true value of p? n= Enter your answer in accordance to the item c) of the question statement

X₁,...,X_n are i.i.d Γ(α₁,β),...,Γ(α_n,β) respectively. Show S = X₁ +···+X_n ∼ Γ(α₁ +···,α_n,β).

The Wechsler IQ test has a normal shape with μ = 100 and σ = 15. Describe the expected characteristics of the distribution of sample means for samples of n = 45. Mean = Standard error Shape = Still speaking of the Wechsler IQ test, if you take a random sample of n = 36 students, what is the probability that the sample mean will be between 96 and 104? Be careful about which formula you use. Still speaking of the Wechsler IQ test, if you take a random sample of n = 144 students, what is the probability that the sample mean will be between 96 and 104?

The heights of all adult American women are normally distributed with a mean of 63.6 inches and a standard deviation of 5 inches. Give the standard (z) score and approximate percentile (from the tables) for women with each of the following heights:

1. 63 inches

2. 60 inches

3. 62.5 inches

Let X have a beta distribution with parameters (a,b). Derive a formula for E(X^r) and from this find the mean and variance.

Researchers examined the bottled-water phenomenon and preference of water types. One hundred fifteen subjects participated in double-blind taste tests of three different bottled water brands (Fiji, Aquafina, and Sam’s Choice) and tap water. Fifteen people preferred the tap water, 29 Aquafina, 47 Fiji, and 24 Sam’s Choice. At the 1% significance level, do the data provide sufficient evidence to conclude that the four different water types are not equally likely in preference?

Complete the table below

Then use (a) the critical value approach and (b) the P-value approach

(Note: Conclusion should be the same no matter the approach used)

For each test,

i) State your null and alternative hypotheses.

ii) What is the significance level?

iii) What is the chi-square test statistic?

iv) What conclusion can you draw based on the data? Write a sentence. Include your test results.

Thirty small communities in Connecticut (population near 10,000 each) gave an average of x = 138.5 reported cases of larceny per year. Assume that σ is known to be 44.5 cases per year. (a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(d) Compare the margins of error for parts (a) through (c). As the confidence levels increase, do the margins of error increase?

As the confidence level increases, the margin of error decreases. As the confidence level increases, the margin of error increases.     As the confidence level increases, the margin of error remains the same.

(e) Compare the lengths of the confidence intervals for parts (a) through (c). As the confidence levels increase, do the confidence intervals increase in length?

As the confidence level increases, the confidence interval remains the same length. As the confidence level increases, the confidence interval decreases in length.     As the confidence level increases, the confidence interval increases in length.

An industrial engineer at an appliance repair company compared a new strategy for dispatching its service technicians to its usual way. The new strategy consisted of using software to solve the vehicle routing problem (a mathematical program) each morning after several service calls came in. The old way was to simply wait for a service request, then send any available technician. Each of the thirty-six technicians was assigned to either the old or new way, so that there were exactly eighteen in each group. The company carried out this experiment over 40 working days, and the average daily mileages for the thirty-six repairpersons are below.

Old

New

a) Make boxplots, probability plots, and run Anderson-Darling and Shapiro-Wilks tests to check for normality. Is the normality assumptions in question Explain.
b) If you think the normality assumptions are violated, you need to run some other kind of test, like a nonparametric test. One reasonable choice would be the Mann-WhitneyWilcoxon rank sum test. Use R to run this test. Provide your code, output, and remarks in your solutions.

A set of X-Y data pairs are

Compute the intercept β₀ regression coefficient for this data.

Out of 500 Respondents in a recent health survey 47 reported a history of diabetes.

a. Estimate the true proportion of people with history of diabetes with 95% confidence.

b. What should the sample size be if the researchers wanted to be accurate to within 2% of the true proportion?

Given an approximately normal distribution with a mean of 175 and a standard deviation of 37.

(a) Draw a normal curve and label 1, 2, and 3 standard deviations on both sides on the mean.

(b) What percent of values are within the interval (138, 212)?

(c) What percent of values are within the interval (101, 249)?

(d) What percent of values are within the interval (64, 286)?

(e) What percent of values outside the interval (138, 212)?

(f) What percent of values are outside the interval (101, 249)?

(g) What percent of values are outside the interval (64, 286)?

A congressional committee of three is to be chosen from a set of 4 Democrats, 4 Republicans, and 3 Independents.

1) find p(all 3 are from different groups)

2) find p(at least 1 democrat)

3) find p(the majority are independents)

a) Use the following data from eight individuals who were measured on variables X and Y, to calculate the requested sums, below.       

a. ΣX        b. Σ(X - Y)         c. ΣX²        d. Σ(X)²

e. (ΣX)²        f. Σ(Y - X)³        g. ΣXY        h. ΣX - ΣY

b) Use the following data from eight individuals who were measured on variables X and Y, to calculate the requested sums, below.

a. ΣX            b. ΣY            c. ΣXY            d. ΣX²

e. (ΣX)²            f. (ΣXY)²        g. (ΣX)(ΣY)        h. Σ(X - 3)

i. Σ(X - 2)(Y - 3)        j. Σ(Y - 3)²