1. A cereal company is interested in determining if there is a difference in the variation of the weights for 24-ounce and 48-ounce boxes of cereal.  A random sample of 18 (eighteen) 24-ounce boxes of cereal produced a sample variance (S₁²) of 0.0049 oz².  A sample of thirty-one (31) 48-ounce boxes of cereal produced a sample variance (S₂²) of 0.006 oz².  Use the sample information to construct a 95% confidence interval estimate for the true population ratio of. The point estimate for the ratio is calculated as  which equals 0.817 (You need to find the lower and upper limits of the confidence interval by completing parts (a) and (b) below).

1. Find the F interval coefficients, the upper F value and the lower F value from the F-table (or from software) for computing a 95% confidence interval estimate of the ratio of the two population variances. Note: the point estimate of the ratio is given as S₁²/ S₂²

F_upper =

F_Lower =

2. Construct the 95% confidence interval for the ratio of the two population variances.

3. Based on the confidence interval calculated in part (b) what would you conclude about the two population variances?

1. The confidence interval includes 1. Conclude that the two population variances are equal
2. The confidence interval includes 1. Conclude that the two population variances are not equal
3. The confidence interval does not include 1. Conclude that the two population variances are equal
4. The confidence interval does not include 1. Conclude that the two population variances are not equal

Directions: Complete each of the following problems. Be sure to show your work in order to receive full or partial credit. Calculators are allowed along with 1 page (1 side) of notes any any necessary statistical tables.

1. (True / False) Estimating parameters and testing hypotheses are two important aspects of descriptive statistics.

2. (True / False) A statistic is calculated from a population and a parameter is calculated from a sample.  

3. (True / False) Descriptive statistics include visual display of data, measures of central tendency, and dispersion.

What type of data (attribute, discrete numerical, continuous numerical) is each of the following variables:

4. ______________________ The manufacturer of your laptop computer

5. ______________________ The number of tickets in a movie theater

Which level of data (nominal, ordinal, interval, ratio) is each of the following variables?

6. ______________________ Your social security number

7. ______________________ Temperature in degrees Celsius

8. 1,000 names are selected from a phone book containing 50,000 people by choosing every 50^th name. Which sampling method is this?

A) Simple random sample.

B) Systematic sample.

C) Stratified sample.

D) Cluster sample.

9.From its 32 regions, the F.A.A. selects 6 regions, and then randomly audits 25 departing commercial flights in each region for compliance with legal fuel and weight requirements. This is an example of

A) simple random sampling.

B) stratified random sampling.

C) cluster sampling.

D) judgment sampling.

10. Suppose we want to estimate vaccination rates among employees in state government, and we know that our target population is 55 percent male and 45 percent female. Our budget only allows a sample size of 200. We randomly sample 110 males and 90 females. This is an example of

A) simple random sampling.

B) stratified random sampling.

C) cluster sampling.

D) judgment sampling.

11. In 2018, the mean per capita expenditures on public libraries for all 50 states was 18 with a standard deviation of 5. The data follow a bell-shaped curve.

      a.    According to the empirical rule, what percentage of the expenditures should fall between 13 and 23? ________________

2. Within what interval should approximately 99.7% of the expenditures fall? _________________

12. Find the standard normal area under the curve for each of the following

Why do you think that some researchers are criticizing significance testing?

Management of the Telemore Company is considering developing and marketing a new product. It is estimated to be twice as likely that the product would prove to be successful as unsuccessful. If it were successful, the expected profit would be $1,500,000. If unsuccessful, the expected loss would be $1,800,000. A marketing survey can be conducted at a cost of $300,000 to predict whether the product would be successful. Past experience with such surveys indicates that successful products have been predicted to be successful 80% of the time, whereas unsuccessful products have been predicted to be unsuccessful 70% of the time. a) (10 points) Find the unconditional probability that the research predicts the product to be successful. Also, find the unconditional probability that the research predicts the product to be unsuccessful. b) (10 points) Find the posterior probabilities of the respective states of nature for each of the two possible predictions from the market survey (your answer should have four probabilities). c) (40 points) Draw the decision tree, including labeling all the decision and outcome nodes and branches, payoffs on all branches, and probabilities on outcome branches. Add up payoffs along each path from the root to a leaf node to obtain the payoff for the leaf node. d) (40 points) Use the backward induction procedure to find the optimal policy to maximize expected payoff. To get full credit, record all the expected payoffs used during the process, use a double dash (||) to block rejected decisions and state the optimal policy in words.

A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a chairperson, secretary, and treasurer. Nobody can hold more than one of these positions.

a. How many selections are there in which Dolph is either a chairperson or he is not an officer?

b. How many selections are there in which Ben is either chairperson or treasurer?

c. How many selections are there in which either Ben is chairperson or Alice is secretary?

The movie industry is a competitive business. The opening weekend gross sales ($ millions), the total gross sales ($ millions), the number of theaters the movie was shown in, and the number of weeks the movie was in release are common variables used to measure the success of a movie. Data on the top 100 grossing movies released in 2016 (Box Office Mojo website) are contained in the attached Excel file. We will use the numerical methods of descriptive statistics discussed in Chapter 3 to create a report of our findings.

1. Find descriptive statistics, including the mean, median, mode, range, standard deviation, and quartiles, for each of the four variables described above. Discuss what the statistics you find tell us about the movie industry.

2. According to Total Gross Sales, what movies if any, should be considered high-performance outliers? Explain how you arrived at your answer mathematically. (Use the method on p. 134 for detecting outliers.) Round any calculations/data to two decimal values.

3. Compute descriptive statistics showing the relationship between total gross sales and each of the other variables. These need to include the covariance and correlation coefficient. Discuss the relationships.

Calculate the sample covariance and also calculate the sample mean and variance for the advertising and sales variables.

David sells ice cream at a local playground. His average daily income is $200 with a standard deviation of $30. Assume David's earnings are normally distributed.

a) David made $180 today. Based on his past earnings, what is the z-score for $180? (Round to no less than two decimal places.

b) What is the probability that David will make $180 or more tomorrow? (Give the proportion correct to four decimal places.)

(a) Two teams, A and B, are playing in the best-of-7 World Series; whoever gets to 4 wins first wins the series. Suppose the home team always has a small advantage, winning each game with probability 0.6 and losing with probability 0.4. Also assume that every game is independent. What is the probability that team A will win the series in exactly 6 games if the series is played in the following format: A–A–B–B–B–A–A, meaning that the first two games are played on team A’s field, followed by three games on team B’s field, and the final two games back on team A’s field?

[Note: Do not use the negative binomial straight up. You will run into trouble, because in this case, the winning probability shifts from one team to the other depending on who has the home field advantage.]

(b) You’re a huge Boston Red Sox fan, and in the current1 best-of-7 World Series they have a probability of p = 0.4 of winning each game. After the Sox lose Game 1, you get so inebriated that you sleep for two days, and miss the next two games. Upon awakening, you rush out to the street and ask the first person you see, “What happened in Games 2 and 3?” “They split them,” comes the reply. Should you be happy? In other words, how do the Sox’s chances of winning look now compared to after Game 1?

The Telektronic Company purchases large shipments of fluorescent bulbs and uses this acceptance-sampling plan: Randomly select and test 20 bulbs, then accept the whole batch if there is only one or none that doesn’t work. If a particular shipment of thousands of bulbs actually has a 4.5% rate of defects, what is the probability that this whole shipment will be accepted? [Assume a binomial probability distribution.]

14% of all Americans live in poverty. If 46 Americans are randomly selected, find the probability that

A. Exactly 7 of them live in poverty.

B. At most 6 of them live in poverty.

C. At least 4 of them live in poverty.

D. Between 4 and 8 (including 4 and 8) of them live in poverty.

The objective of a study was to see whether a recorded phone would be more effective than a mailed flyer in getting voters to support a certain candidate. The study assumes a significance level of α = 0.05.

The hypotheses are:

H0: p(voted to support candidate with flyer) – p(voted to support candidate with recorded phone call) = 0, and

HA: p(voted to support candidate with flyer) – p(voted to support candidate with recorded phone call) > 0.

(a) Explain what the p-value (0.027) indicates with respect to the observed sample statistic (and other, more extreme values of that statistic). Name the sample statistic involved as well as the p-value, and use the appropriate mathematical notation. (1-2 sentences.)

(b) Write a specific statement about your interpretation of the null hypothesis, given the p-value and the specified level of significance. Be sure to cite the p-value. Does the sample evidence available support the idea that phone calls are more effective than flyers? Explain.

(c) In the conclusion for (a) & (b), which type of error are we possibly making: Type I or Type II? Explain what this error means in this context.

(d) What if the p-value for the statistical test were actually 0.18 (and not 0.027)? Explain what the p-value (0.18) indicates with respect to the observed sample statistic (and other, more extreme values of that statistic). Name the sample statistic involved, report the p-value, and use the appropriate mathematical notation. (1-2 sentences.)

(e) Write a specific statement about your interpretation of the null hypothesis, given the p-value from (d), 0.18, and the specified level of significance. Be sure to cite the p-value and α. Does the sample evidence available support the idea that phone calls are more effective than flyers? Explain.

(f) In the conclusion from (e), which type of error are we possibly making: Type I or Type II? Describe what this error means in this context.