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Chapter 7: Q. 57 (page 479)

Bottling cola 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 渭=298mland standard deviation 蟽=3ml.

a. What is the probability that a randomly selected bottle contains less than 295ml?

b. What is the probability that the mean contents of six randomly selected bottles is less than 295ml?

Short Answer

Expert verified

(a). the probability that a randomly selected bottle contains less than 295mlis 15.87%

(b). the probability that the mean contents of six randomly selected bottles is less than is0.71%

Step by step solution

01

Part(a) Step 1: Given information 

Given,

渭=298蟽=3x=295

Use the following formulae

z=x-渭蟽

02

Part(a) Step 2: Calculation

The z-score is

z=x-渭蟽=295-2983=-1.00

The likelihood of associating using the normal probability P(Z<-1.00)is given in the row starting with -1.0and in the column starting with .00of the standard normal probability

P(x<295)=P(Z<-1.00)=0.1587=15.87%

03

Part(b) step 1: Given information 

Given,

渭=298蟽=3n=6x=295
04

Part(b) Step 2: Calculation

Because the population distribution is normal, so is the sampling distribution of the sample mean x炉.

The z-score is

z=x-渭x炉炉蟽x炉炉=x炉-渭dn=295-2983n炉=-2.45

The associating probability using the normal probability P(Z<-2.45)is given in the row starting with -2.4and in the column starting with .05of the standard normal probability

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Most popular questions from this chapter

The manufacturer of a certain brand of aluminum foil claims that the amount of foil on each roll follows a Normal distribution with a mean of 250 square feet (ft2 ) and a standard deviation of 2 ft2 . To test this claim, a restaurant randomly selects 10 rolls of this aluminum foil and carefully measures the mean area to bex=249.6ft2.

a. Find the probability that the sample mean area is 249.6ft2or less if the manufacturer鈥檚 claim is true.

b. Based on your answer to part (a), is there convincing evidence that the company is overstating the average area of its aluminum foil rolls?

Refer to the small population of 5students in the table.

Sample means List all 10 possible SRSs of size n=2, calculate the

mean quiz score for each sample, and display the sampling distribution of the sample

mean on a dotplot.

According to government data, 22% of American children under the age of 6 live in households with incomes less than the official poverty level. A study of learning in early childhood chooses an SRS of 300 children from one state and finds that p鈭�p^=0.29.

a. Find the probability that at least 29% of the sample are from poverty-level households, assuming that 22% of all children under the age of 6 in this state live in poverty-level households.

b. Based on your answer to part (a), is there convincing evidence that the percentage of children under the age of 6 living in households with incomes less than the official poverty level in this state is greater than the national value of 22%? Explain your reasoning.

An insurance company claims that in the entire population of homeowners, the mean annual loss from fire is and the standard deviation of the loss is 蟽=\(5000.The distribution of losses is strongly right-skewed: many policies have \)0loss, but a few have large losses. The company hopes to sell 1000 of these policies for \(300each.

a. Assuming that the company鈥檚 claim is true, what is the probability that the mean loss from fire is greater than \)300for an SRS of 1000 homeowners?

b. 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?

Dead battery? A car company claims that the lifetime of its batteries varies from car to car according to a Normal distribution with mean 渭=48months and standard deviation 蟽=8.2months. A consumer organization installs this type of battery in an SRS of 8 cars and calculates x炉=42.2months.

a. Find the probability that the sample mean lifetime is 42.2 months or less if the company's claim is true.

b. Based on your answer to part (a), is there convincing evidence that the company is overstating the average lifetime of its batteries?
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