The chefs at a local pizza chain, strive to maintain the suggested size of their 16-inch pizzas. Despite their best efforts, they are unable to make every pizza exactly 16 inches in diameter. The manager has determined that the size of the pizzas is normally distributed with a mean of 16 inches and a standard deviation of 0.8 inch.

a. What are the expected value and the standard error of the sample mean derived from a random sample of 2 pizzas?

b. What are the expected value and the standard error of the sample mean derived from a random sample of 4 pizzas?

c. Compare the expected value and the standard error of the sample mean with those of an individual pizza.

If E(X) = 5, V(X) = 3 and Y = 5 - X/3 - X, what is mean of Y?

The table represents the answers of 80 respondents to the survey “How much sports
trainings do you have every year?” carried out among college students. Construct a stem-and-leaf
diagram for the data. Calculate the median and quartiles of these data.

129 157 154 191 192 142 188 126 128 180
190 166 157 147 155 154 200 128 167 143
131 156 153 168 149 144 155 188 149 142
160 149 184 187 169 161 157 134 122 173
188 183 178 148 135 188 187 166 121 177
169 182 158 169 146 173 133 189 183 143
148 121 181 145 189 120 122 189 146 190
128 142 189 131 199 182 197 148 157 140

The article “Determination of Carboxyhemoglobin Levels and Health Effects on Officers Working at the Istanbul Bosphorus Bridge” (G. Kocasoy and H. Yalin, Journal of Environmental Science and Health, 2004:1129–1139) presents assessments of health outcomes of people working in an environment with high levels of carbon monoxide (CO). Following are the numbers of workers reporting various symptoms, categorized by work shift. The numbers were read from a graph.

Can you conclude that the proportions of workers with the various symptoms differ among the shifts?

(a) State the appropriate null hypothesis.

(b) Compute the expected values under the null hypothesis.

(c) Compute the value of the chi-square statistic.

(d) Find the p-value. What do you conclude?

Suppose two Gaussian R.V.'s X and Y follow (X,Y)~N(1, 0; 1, 4, 0.5), and W=3X+Y. Find the joint PDF between W and Y, fwy(w,y) and find E{W|Y=y}.

What is the critical value of the correlation coefficient needed for rejection of the null hypothesis if your degrees of freedom is 30 and you are conducting a two-tailed test using the .05 level of significance?

The numbers racket is a well‑entrenched illegal gambling operation in most large cities. One version works as follows: you choose one of the 1000 three‑digit numbers 000 to 999 and pay your local numbers runner a dollar to enter your bet. Each day, one three‑digit number is chosen at random and pays off $600 . The law of large numbers tells us what happens in the long run. Like many games of chance, the numbers racket has outcomes that vary considerably—one three‑digit number wins $600 and all others win nothing—that gamblers never reach “the long run.” Even after many bets, their average winnings may not be close to the mean. For the numbers racket, the mean payout for single bets is $0.60 ( 60 cents) and the standard deviation of payouts is about $18.96 . If Joe plays 350 days a year for 40 years, he makes 14,000 bets. Unlike Joe, the operators of the numbers racket can rely on the law of large numbers. It is said that the New York City mobster Casper Holstein took as many as 25,000 bets per day in the Prohibition era. That's 150,000 bets in a week if he takes Sunday off. Casper's mean winnings per bet are $0.40 (he pays out 60 cents of each dollar bet to people like Joe and keeps the other 40 cents). His standard deviation for single bets is about $18.96 , the same as Joe's.

(a) What is the mean of Casper's average winnings ?¯ on his 150,000 bets? (Enter your answer as dollars rounded to two decimal places.)

mean of average winnings=

What is the standard deviation of Casper's average winnings ?¯ on his 150,000 bets? (Enter your answer as dollars rounded to three decimal places.)

standard deviation=$

(b) According to the central limit theorem, what is the approximate probability that Casper's average winnings per bet are between $0.30 and $0.50 ? (Enter your answer rounded to four decimal places.)

approximate probability=

One way of making a series stationary is?

Overview

The SPSS output tables below are all based on a larger study of product assessments for an ecologically friendly engine oil. Participants saw the packaging of an ecologically friendly engine oil on their computer screen and were asked several questions regarding how they perceived this product. The tables below focus on only a few of the variables used in the study.

Coding

Gender was coded as 1=female and 2=male

PWOP_M_S refers to perceived warmth of product. Participants were asked to which extent they perceived the product as “warm.” The assumption is that the perception of the color of the product’s packaging influences this assessment. For this Chapter 11 exercise, I performed a median split on the variable. Thus, the variable is now dichotomous with 1=low perceived warmth and 2=high perceived warmth.

FLUEN_M_S refers to processing fluency. Participants were asked to which extent they perceived the product and its packaging as easy to process, well organized, logical, etc. For this Chapter 11 exercise, I performed a median split on the variable. Thus, the variable is now dichotomous with 1=low fluency and 2=high fluency.

A_PI refers to purchase intentions. This variable is continuous on a scale from 1 to 7. Higher values represent higher purchase intentions.

Note 1: It may be true that you are not fully familiar with the constructs and do not know much about the study context, but you can nevertheless interpret the results provided in a table from a statistical point of view.

Note 2: When responding to the questions, please provide your answers in a way which will encourage the reader to believe that you understood the logic of these statistical tests. For example, it is helpful to point out which numbers in the tables are important, and what the meaning of these numbers is.

Question 11.1

The following two tables are equivalent to Exhibit 11.10 in your textbook (Crosstab Chi-Square example).

Please provide an interpretation of the two tables. What are the insights we can obtain from the SPSS output shown below? What is the meaning of the “count” vs. the “expected count” information in the table? (explain the logic with an example from the table). What is the logic of the Chi Square tests (and specifically, what is the meaning of the numbers shown in the “Asymp. Sig” column?

The Cheebles cookie factory changed their recipe. The inspectors took a sample of the new cookies and found that the sample was 42 grams with a standard deviation of 4 grams. the Cheebles CEO specially asked the inspectors to use these statistics to find the lower and upper boundary weighs of 50% of their cookies. What are the Z values of the limits of the limits of the area covering the middle half of the area under the normal curve that the inspectors would use to find this information for the CEO of Cheebles?

Suppose that the Gross Domestic Product (GDP) in the US is denoted as G_t. Let the quarterly data of G_t from 1980:1 to 2015:1 is non-stationary but the first difference of G_t, denoted as DG_t, is stationary. Assume that a researcher identified the following AR model for DG_t:

DG_t = Alpha₀ + Alpha₁DG_t-1 + Alpha₃DG_t-3 + Alpha₄DG_t-4 + e_t

Suppose the estimated results of the above AR model are as follows:

DG_t = 6.09 + 0.18DG_t-1 + 0.12DG_t-3 + 0.05DG_t-4

(a) Based on what criteria of the coefficients of auto-correlation function (ACF) and the coefficients of partial auto-correlation function (PACF), the researcher identified the above AR model? Explain. Also explain why the researcher chose 1, 3, and 4 lags.

(b) Forecast the GDP for 2015:2, assuming that the GDP in 2015:1, 2014:4, 2014:3, 2014:2, 2014:1, 2013:4, 2013:3, and 2013:2 respectively are: 16264.1, 16294.7, 16205.6, 16010.4, 15831.7, 15916.2,15779.9, and 15606.6.

(c) Explain how to conduct a diagnostic test to check if the researcher has identified the correct AR model. (Hint: Step 3 of Box-Jenkin’s Method)

Please show work.

A machine is designed to fill 16-ounce bottles of shampoo. When the machine is working properly, the amount poured into the bottles follows a normal distribution with mean 16.05 ounces with a standard deviation of .1 ounces. If four bottles are randomly selected each hour and the number of ounces in each bottle is measured, then 95% of the means calculated should occur in what interval? Hint: the standard deviation rule says that 95% of the observations are within how many standard deviations away from the mean? Round answers to four decimal places

.

1.) Consideration of Causation:

(i) Describe a realistic, but a hypothetical example of a situation in public administration where a novice researcher might incorrectly think that a clear correlation between two variables implies that one caused the other.?

(ii) Then, discuss the correct research-based approach to the analysis of your hypothetical example. In other words, how should the relationship between the two variables be accurately described or approached by a seasoned researcher?

2. Earlier in this chapter, we discussed that iron supplements are popular in part because they raise oxygen levels in our bodies, and increased oxygen levels help us feel more energetic. We also said that consuming an iron supplement with a drink high in Vitamin C enhances the effects of the iron supplement. A researcher has a sample of 40 people randomly assigned to two factors: iron supplement and Vitamin C consumption. Specifically, people were randomly assigned either to take an iron supplement or not to take an iron supplement. In addition, people were also randomly assigned either to consume a glass of orange juice (which is high in Vitamin C) or not to consume a glass of orange juice. The researcher uses a pulse oximetry to measure oxygen levels as the dependent variable (to keep the calculations simple, we are using intentionally hypothetical numbers for oxygen levels).The data is presented below. Carry out the analysis of variance using SPSS and determine whether there are any main effects or an interaction. Make a line graph of the results and interpret the pattern of the results.

The time until the light in Bob's office fails is exponentially distributed with mean 2 hours. The time until the computer crashes in Bob's office is exponentially distributed with mean 3 hours. Failure and crash times are independent.

(a) Find the probability that neither the light nor computer fail in the next 2 hours

(b) Find the probability that the computer crashes at least 1 hour after the light fails.