The number of times a machine broke down each week was observed over a period of 100 weeks and recorded in the accompanying table. It was found that the average number of breakdowns per week over this period was 2.1. Test the null hypothesis that the population distribution of breakdown is Poisson.

$$\begin{array}{lcccccc} \hline \text { Number of breakdowns } & 0 & 1 & 2 & 3 & 4 & 5 \text { or more } \\ \hline \text { Number of weeks } & 10 & 24 & 32 & 23 & 6 & 5 \\ \hline \end{array}$$

The discrete random variable X has the probability distribution P(X = x) = ln kx for x = 1, 2, 3, 4 . Find the exact value of k.

A random sample of statistics professors was asked to complete a survey including questions on curriculum content, computer integration, and software preferences. Of the 250 responses, 100 professors indicated that they preferred software package M and 80 preferred software package E, whereas the remainder were evenly split between preference for software package S and software package P. Do the data indicate that professors have a preference for any of these software packages?

A random sample of 150 residents in one community was asked to indicate their first preference for one of three television stations that air the 5 p.m. news. The results obtained are shown in the following table. Test the null hypothesis that for this population their first preferences are evenly distributed over the three stations.

$$\begin{array}{lccc} \hline \text { Station } & \text { A } & \text { B } & \text { C } \\ \hline \text { Number of first preferences } & 47 & 42 & 61 \\ \hline \end{array}$$

Suppose a computer chip manufacturer rejects 2% of the chips produced because they fail presale testing. a) What’s the probability that the fifth chip you test is the first bad one you find? b) What’s the probability you find a bad one within the first 10 you examine?

Let x be a Poisson random variable with mean $\mu$=2. Calculate these probabilities: a. P(x=0) b. P(x=1) c. P(x$\gt$1) d. P(x=5)

Do these situations involve Bernoulli trials? Explain. You are rolling 5 dice and need to get at least two 6's to win the game.

Given independent random variables with means and standard deviations as shown,

$$\begin{array}{c|c|c} & \text { Mean } & \text { SD } \\ \hline \boldsymbol{X} & 80& 12 \\ \boldsymbol{Y} & 12 & 3 \end{array}$$

Find the mean and standard deviation of $2Y+20$.

Abalones are edible sea snails that include over 100 species. A researcher is working with a model that uses the number of rings in an abalone's shell to predict its age. He finds an observation that he believes has been miscalculated. After deleting this outlier, he redoes the calculation. Does it appear that this outlier was exerting very much influence?

$$\begin{array}{l} \text{Before:}\\ \text{Dependent variable is Age}\\ \text{R-squared = 67.5\\\%}\\ \\ \begin{array}{lr} \text{Variable} & \text{Coefficient}\\ \text{Intercept} & 1.736\\ \text{Rings} & 0.45 \end{array} \end{array}$$

$$\begin{array}{l} \text{After:}\\ \text{Dependent variable is Age}\\ \text{R-squared = 83.9\\\%}\\ \\ \begin{array}{lr} \text{Variable} & \text{Coefficient}\\ \text{Intercept} & 1.56\\ \text{Rings} & 1.13 \end{array} \end{array}$$

According to the 2010 Census, 11.4% of all housing units in the United States were vacant. A county supervisor wonders if her county is different from this. She randomly selects 850 housing units in her county and finds that 129 of the housing units are vacant. a) State the hypotheses. b) Name the model and check appropriate conditions for a hypothesis test. c) Draw and label a sketch, and then calculate the test statistic and P-value. d) State your conclusion.

According to the 2010 Census, 16% of the people in the United States are of Hispanic or Latino origin. One county supervisor believes her county has a different proportion of Hispanic people than the nation as a whole. She looks at their most recent survey data, which was a random sample of 437 county residents, and found that 44 of those surveyed are of Hispanic origin. a) State the hypotheses. b) Name the model and check appropriate conditions for a hypothesis test. c) Draw and label a sketch, and then calculate the test statistic and P-value. d) State your conclusion.

A coffee shop tracks sales and has observed the distribution in the following table. What is the standard deviation ?

$$\begin{array}{l|c|c|c|c|c} \text { \# of Sales } & 145 & 150 & 155 & 160 & 170 \\ \hline \text { Probability } & 0.15 & 0.22 & 0.37 & 0.19 & 0.07 \end{array}$$

A manufacturer ships toaster s in cartons of 20 In each carton , they estimate a 5% chance that one of the toasters will need to be sent back for minor repairs. What is the probability that in a carton , there will be exactly 3 toasters that need repair?

The technology committee of the earlier exercise from the previous chapter wants to perform a test to see if the mean amount of time students are spending in the lab has increased from 55 minutes. Here are the data from a random sample of 12 students.

$$\begin{array}{c} \begin{array}{c} \text{Time} \end{array} \\ \begin{array}{l|r} \hline 52 & 74 \\ 57 & 53 \\ 54 & 136 \\ 76 & 73 \\ 62 & 8 \\ 52 & 62 \end{array} \end{array}$$

What do you conclude about the claim? If there are outliers, perform the test with and without the outliers present.