Let x be a random variable representing percentage change in neighborhood population in the past few years, and let y be a random variable representing crime rate (crimes per 1000 population.) A random sample of six Denver neighborhoods gave the following information:x 29 2 11 17 7 6y 173 35 132 127 69 53Σx=72, Σy=589, Σx^2=1340, Σy^2=72,277,Σxy=9499a) draw a scatter diagram for the datab) find x bar, y bar, b and the equation of the least-squares line. Plot the line on the scatter diagram of part (a).c) Find the sample correlation coefficient r and the coefficient of determination. What percentage of the variation in y is explained by the least-squares model?d) Test the claim that the population correlation coefficient p is not zero at the 1% level of significance.e) For a neighborhood with x= 12% change in population in the past few years, predict the change in the crime rate (per 1000 residents)f) verify that Se= 22.5908g) Find a 80% confidence interval for the change in crime rate when the percentage change in population is x= 12% h) Test the claim that the slope B of the population least-squares line is not zero at the 1% level of significance.I) Find an 80% confidence interval for B and interpret its meaningshow all work!

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Let X be exponentially distributed with parameter λ > 0. Find the PMF of Y = ⌈X⌉, where, given a real number x, ⌈x⌉ denotes the rounding of x to the nearest integer whose value is greater or equal to x. Identify the distribution of Y .

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1. Let X be the number of Chargers (NFL team in Los Angeles) fans observed in a random sample of n = 30 graduate students at Penn State’s University Park campus. Assume that X ∼ Bin(30; π). The true proportion π is unknown, but it is likely to be small (especially in comparison to the potential number of Pittsburgh Steelers fans).
2. (a) Assume for now that π = 0.04. Find the probability that X = 0.
3. (b) The classic (Wald) approximate 95% confidence interval for π is
4. √
5. πˆ(1 − πˆ) n
6. where πˆ = X/n. When does this interval become degenerate (i.e., when the lower and upper bounds are the same)? If the true π were actually 0.04, could this interval actually cover the true parameter 95% of the time? Why or why not? Hint: based on part (a), how often would the interval become degenerate at zero?
7. (c) Suppose that we observe two Chargers fans in the sample. Plot the log-likelihood func- tion for π over a range of values from 0.01 to 0.20. Find the ML estimate, and calculate the approximate 95% confidence interval for π based on the formula above.
8. (d) Now find an approximate 95% confidence interval based on the likelihood ratio method. That is, find the range of null values that the likelihood ratio test would fail to reject.
9. (e) In their 1998 paper, Agresti and Coull propose using the Wald interval but with two pseudo successes and two pseudo failures first added to the sample. Compute the 95% interval based on this modification, and compare it with the other intervals calculated above. 

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Let (X, dX) and (Y, dY ) be metric spaces, and let f : X → Y be continuous. Define the distance d on the product space X × Y as in class, so that (X × Y, d) is a metric space. Show that the graph Γf of f, defined by Γf = {(x, y) ∈ X × Y | f(x) = y}, is a closed subset of X × Y .

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xx $27$8

(a) Consider a random sample of n = 40 customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution of x, the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the x distribution?

The sampling distribution of x is not normal.

The sampling distribution of x is approximately normal with mean μx = 27 and standard error σx = $8.    

The sampling distribution of x is approximately normal with mean μx = 27 and standard error σx = $0.20.

The sampling distribution of x is approximately normal with mean μx = 27 and standard error σx = $1.26.

Is it necessary to make any assumption about the x distribution? Explain your answer.

It is not necessary to make any assumption about the x distribution because μ is large.

It is necessary to assume that x has an approximately normal distribution.    

It is necessary to assume that x has a large distribution.

It is not necessary to make any assumption about the x distribution because n is large.

(b) What is the probability that x is between $24 and $30? (Round your answer to four decimal places.)

(c) Let us assume that x has a distribution that is approximately normal. What is the probability that x is between $24 and $30? (Round your answer to four decimal places.)

(d) In part (b), we used x, the average amount spent, computed for 40 customers. In part (c), we used x, the amount spent by only one customer. The answers to parts (b) and (c) are very different. Why would this happen?

The standard deviation is smaller for the x distribution than it is for the x distribution.

The sample size is smaller for the x distribution than it is for the x distribution.    

The standard deviation is larger for the x distribution than it is for the x distribution.

The mean is larger for the x distribution than it is for the x distribution.

The x distribution is approximately normal while the x distribution is not normal.

In this example, x is a much more predictable or reliable statistic than x. Consider that almost all marketing strategies and sales pitches are designed for the average customer and not the individual customer. How does the central limit theorem tell us that the average customer is much more predictable than the individual customer?

The central limit theorem tells us that small sample sizes have small standard deviations on average. Thus, the average customer is more predictable than the individual customer.

The central limit theorem tells us that the standard deviation of the sample mean is much smaller than the population standard deviation. Thus, the average customer is more predictable than the individual customer.  

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