1. Determine each of the following is true or false? If false, provide a counterexample.
(a) Let X be a continuous random variable which has the pdf fX. Then, for each x, 0 ≤ fX(x) ≤ 1.

(b) Any two independent random variables have ρXY = 0.

(c) Let X and Y be random variables such that E[XY ] = E[X]E[Y ]. Then, X and Y are independent.

2. Ann plays a game with Bob. Ann draws a number X1 ∼ U(0,1) and Bob draws a number X2 ∼ U(0,1). Assume X1 and X2 are independent.
(a) Calculate the conditional probability of Ann winning the game given Ann draws x1 ∈ [0,1].

(b) Calculate the probability of Ann winning the game. Hint: this is equal to P(X1 > X2). You may calculate this directly using integration; An alternative way is to use a geometric intuition. If x-axis represents x1 and y-axis represents x2, what does the set of (x1,x2) such that x1 > x2 look like? What is the size of this set?

A newly installed automatic gate system was being tested to see if the number of failures in 1,000 entry attempts was the same as the number of failures in 1,000 exit attempts. A random sample of eight delivery trucks was selected for data collection. Do these sample results show that there is a significant difference between entry and exit gate failures? Use α = 0.05.

(c) Find the critical value t_crit for α = 0.05. (Round your answer to 3 decimal places. A negative value should be indicated by a minus sign.)



(d) Find the p-value. (Round your answer to 4 decimal places.)

  

Problem 1. A set of 25 measurements consists of the values
3 6 7 15 12
6 8 4 5 6
5 12 1 3 3
7 5 10 8 3
9 2 6 1 5
(a) Construct a relative frequency histogram to describe the data. (10 pts) (b) Find: the
median m (5 pts), the sample mean ¯x (5 pts), the sample standard deviation s (5 pts), the
upper and lower quartiles (10 pts), the 37-th percentile (5 pts). (c) Apply Tchebisheff’s
Theorem in order to estimate the proportion of measurements that lie in the interval
[¯x − (1.5)s, ¯x + (1.5)s] (10 pts).

• First, obtain a set of real-world numeric data. You should have 25 to 50 entries in your data set. You can collect your own data or use data from an online source.
• In your initial post, list your data and calculate the five-number summary. Then, pose a problem that involves an analysis of the data. Do not provide a solution. Instead, be sure that you include enough relevant information so that your classmates can propose their solutions in their responses.

• An example of a question may be;

• A cell phone company may ask whether most people send 50 or more text messages per week. To analyze this question, you could ask 30 people how many text messages they sent during the past week. If another student asks 40 other people, do you think that the two of you will have the same conclusion?

An investigation of the effectiveness of a training program to improve customer relationships included a pre-training and post-training customer survey, 12 customers were randomly selected to score the customer relationships both before and after the training. The differences of scores are calculated as the post-training survey score minus the pre-training survey score. The sample mean difference of scores is 0.5. The test statistic is 1.03 for testing whether the population mean difference µD ≤ 0.

11. Based on the above information, the sample standard deviation of the differences of scores is .

A. 1.49

B. 1.53

C. 1.48

D. 1.68

12. Suppose that we want to test whether the training program improved the customer relationships, the null hypothesis is µD ≤ 0. Which of the following is correct?

A. At the 0.05 significance level, we reject the null, the training program improved customer relationships.

B. At the 0.05 significance level, we do not reject the null, the training program did not improve customer relationships.

C. At the 0.10 significance level, we do not reject the null, the training program did not improve customer relationships.

D. At the 0.10 significance level, we reject the null because the test statistic is greater than the critical value 1.645.

A local retail business wishes to determine if there is a difference in the mean preferred indoor temperature between men and women. Assume that the population standard deviations of preferred temperatures are equal between men and women. Two independent random samples are collected. Sample statistics of preferred temperatures are reported in the table below.

Sample size Mean (in Celsius) Standard deviation

Men (Group 1) 25 22.5 1.2

Women (Group 2) 22 20.1 1.4

Suppose that we perform a two-tailed test of the difference of population means, which of the following is the correct conclusion?

3. Consider the following statements about the F distribution:

(i). One characteristic of the F distribution is that F statistics cannot be negative.

(ii). There is one unique F distribution for a F-statistic with 29 degrees of freedom in the numerator and 28 degrees of freedom in the denominator.

(iii). The F distribution’s curve is symmetric around its peak when the degrees of freedom for the numerator and for the denominator are equal.

Which of the following is true?

TechX have received complaints of excessive packaging of one of their products.  To test whether new minimalist packaging is preferred and whether the proportion of people who prefer the new packaging is different for different age groups, they have conducted a market test of the new packaging.  The results are given in the table below:

(a) At a 5 % significance level is the proportion of people who prefer the new packaging the same for all age groups?  Set up and test the appropriate hypotheses.

(b) Suppose that a person is chosen at random from the people in the table above.  If you know that the person chosen is < 20, what is the probability that he/she prefers the new packaging?

A local retail business wishes to determine if there is a difference in the mean preferred indoor temperature between men and women. Assume that the population standard deviations of preferred temperatures are equal between men and women. Two independent random samples are collected. Sample statistics of preferred temperatures are reported in the table below.

Sample size Mean (in Celsius) Standard deviation

Men (Group 1) 25 22.5 1.2

Women (Group 2) 22   20.1 1.4

In testing whether two population means are different, the value of test statistic is .

2) I am about to choose an integer, at random, from amongst the following positive integers: {1,2,3,4, …., 25, 26, 27}

a) is this an example of discrete or continuous uniform probability distribution? Please explain

b) what is the probability that I coincidentally happen to choose integer “23”? show work

c) please depict this probability distribution in some appropriate manner.

d) please determine the probability that I choose: either an even integer, or an integer which is at least 20? Recall: PROB(A or B)+PROB(B)-PROB(A&B). show work

e) please determine the mean value for this probability function. Show work

f) please determine the standard deviation for this probability function. Show work

1-

Find the z-score for which 70% of the distribution's area lies to its right.

A. - 0.98

B. - 0.47

C. - 0.81

D. - 0.53

2-

Find the z-scores for which 98% of the distribution's area lies between - z and z.

A. (- 2.33,2.33)

B.( - 1.645,1.645)

C.( - 0.99,0.99)

D.(- 1.96,1.96)

3- Assume that the heights of women are normally distributed with a mean of 62.4 inches and a standard deviation of 2.5 inches. Find Q 3, the third quartile that separates the bottom 75% from the top 25%.

A. 60.7

B. 65.6

C. 65.3

D.64.1

Of 150 adults who tried a new cranberry-flavored peppermint patty, 90 rated it excellent. Of 250 children sampled, 190 rated it excellent. Use the 0.03 level of significance. Regarding whether there is a significant difference in the proportion of adults and the proportion of children who rate the new flavor as excellent, which of the following is correct?

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 40.3cases 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 increases.

(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 increases in length.

A worker is engaged in the sale of programmed business intelligence and has to choose Among the following options:

Monthly base salary of $ 3,000 plus 5% commission on sales made during the month.

Monthly salary of $ 1,500 plus 8% commission on sales made during the month.

Monthly base salary of $ 3,500 plus 2.3% commission on sales made during the month.

10% commission on sales made during the month.

Each programmed business intelligence has a value of $ 5,000.

Solve the case by answering the next questions:

What option would you recommend to this worker?

Why?

Show all the calculations to arrive at your recommendation and also present them in graph.

In testing a certain kind or truck tire over rugged​ terrain, it is found that 40​% of the trucks fail to complete the test run without a blowout. Of the next 12 trucks​ tested, find the probability that​ (a) from 2 to 6 have​ blowouts, (b) fewer than 4 have​ blowouts, and​ (c) more than 5 have blowouts.