An insurance company claims that in the entire population of homeowners the mean annual loss from fire is $\mu=\$ 250$ and the standard deviation of the loss is $\sigma=\$ 5000.$ The distribution of losses is strongly right-skewed: many policies have $0 loss, but a few have large losses. The company hopes to sell 1000 of these policies for$300 each. If the company wants to be 90% certain that the mean loss from fire in an SRS of 1000 homeowners is less than the amount it charges for the policy, how much should the company charge?

The insurance company in the entire population of homeowners, the mean loss from fire is $$ \mu = $250 $$ and the standard deviation of the loss is $$ \sigma = $300 $$ . The distribution of losses is strongly right-skewed: many policies have $0 loss, but a few have large losses. If the company sells 10,000 policies, what is the approximate probability that the average loss will be greater than$260? Show your method.

An insurance company claims that in the entire population of homeowners, the mean annual loss from fire is $\mu=\$ 250$ with a standard deviation of $\sigma=\$ 5000.$ The distribution of losses is strongly right-skewed: Many policies have $0 loss, but a few have large losses. Describe the shape of the sampling distribution of$$$\bar{x}$ for SRSs of size n = 15 from the population of homeowners. Justify your answer.

A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 milliliters (ml). In fact, the contents vary according to a Normal distribution with mean $\mu=298\ \mathrm{ml}$ and standard deviation $\sigma=3\ \mathrm{ml}.$ What is the probability that the mean contents of six randomly selected bottles is less than 295 ml?

A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 milliliters (ml). In fact, the contents vary according to a Normal distribution with mean $\mu=298\ \mathrm{ml}$ and standard deviation $\sigma=3\ \mathrm{ml}.$ What is the probability that a randomly selected bottle contains less than 295 ml?

An insurance company wants to design a homeowner’s policy for mid-priced homes. From data compiled by the company, it is known that the annual claim amount, X, in thousands of dollars, per homeowner is a random variable with the following probability distribution.

$$\begin{array}{c|ccccc} \hline x & 0 & 10 & 50 & 100 & 200 \\ \hline P(X=x) & 0.95 & 0.045 & 0.004 & 0.0009 & 0.0001 \\ \hline \end{array}$$

a. Determine the expected annual claim amount per homeowner. b. How much should the insurance company charge for the annual premium if it wants to average a net profit of $50 per policy?

A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 3090 milliliters (ml). In fact, the contents vary according to a Normal distribution with mean

$$\mu = 298 ml$$

and standard deviation

$$\sigma = 3 ml$$

. What is the probability that the mean contents of the bottles in a six-pack is less than 295 ml? Show your work.

David's iPod has about 10,000 songs. The distribution of the play times for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds. Describe the shape of the sampling distribution of $\bar{x}$ for SRSs of size n=100 from the population of songs on David's iPod. Justify your answer.

David's iPod has about 10,000 songs. The distribution of the play times for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds. Suppose we choose an SRS of 10 songs from this population and calculate the mean play time

$$\overline{x}$$

of these songs. What are the mean and the standard deviation of the sampling distribution of

$$\overline{x}?$$

Explain.

David's iPod has about 10,000 songs. The distribution of the play times for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds. Describe the shape of the sampling distribution of $\bar{x}$ for SRSs of size n=5 from the population of songs on David's iPod. Justify your answer.

Suppose that a simple random sample is taken from a finite population in which each member is classified as either having or not having a specified attribute. Fill in the following blanks. a. If sampling is with replacement, the probability distribution of the number of members sampled that have the specified attribute is a ____ distribution. b. If sampling is without replacement, the probability distribution of the number of members sampled that have the specified attribute is a ____ distribution. c. If sampling is without replacement and the sample size does not exceed ____% of the population size, the—probability distribution of the number of members sampled that have the specified attribute can be approximated by a _____ distribution.

An insurance company is budgeting the premium rates for the following year. The company decided the rates of the year before based on the average amount they paid as insurance claims, $1000. However, the manager believes that the actual amount of claims that they pay is higher. Thus, they should increase the insurance premium. The manager took a sample of 50 claims, and found the average mean claim of$1,200, with the standard deviation of $500.

a. At 0.05 level of significance, use the six step critical value approach to assist the manager in taking the decision if the average insurance claim of the population is more than$1,000.

b. At the 0.05 level of significance, use the five-step p-value approach to assist the manager in taking the decision if the average insurance claim of the population is more than $1,000.

c. Interpret the meaning of the p-value in (b).

d. Compare your conclusions in (a) and (b).

At a traveling carnival, a popular game is called the "Cash Grab." In this game, participants step into a sealed booth, a powerful fan turns on, and dollar bills are dropped from the ceiling. A customer has 30 seconds to grab as much cash as possible while the dollar bills swirl around. Over time, the operators of the game have determined that the mean amount grabbed is $13 with standard deviation of$9. They charge $15 to play the game and expect to have 40 customers at their next carnival. What is the probability that an SRS of 40 customers grab an average of$15 or more?

A car company claims that the lifetime of its batteries varies from car to car according to a Normal distribution with mean $\mu=48$ months and standard deviation $\sigma=8.2$ months. A consumer organization installs this type of battery in an SRS of 8 cars and calculates $\bar{x}=42.2$ months. Find the probability that the sample mean lifetime is 42.2 months or less if the company's claim is true.