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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

A researcher initially plans to take an SRS of size 160 from a certain population and calculate the sample mean x-x炉. Later, the researcher decides to increase the sample size so that the standard deviation of the sampling distribution of x-x炉will be half as big as when using a sample size of 160 . What sample size should the researcher use?

a. 40

b. 80

c. 320

d. 640

e. There is not enough information to determine the sample size.

Finch beaks One dimension of bird beaks is "depth"-the height of the beak where it arises from the bird's head. During a research study on one island in the Galapagos archipelago, the beak depth of all Medium Ground Finches on the island was found to be Normally distributed with mean 渭=9.5鈥�millimeters(mm)and standard deviation 蟽=1.0mm.

a. Choose an SRS of 5 Medium Ground Finches from this population. Describe the sampling distribution of x炉.

b. Find the probability that x炉estimates 渭within 卤0.5mm. (This is the probability that x炉takes a value between 9 and 10 mm.

c. Choose an SRS of 50 Medium Ground Finches from this population. Now what is the probability that x炉falls within 卤0.05mmof 渭? In what sense is the larger sample "better"?

78 refer to the following setting. In the language of government statistics, you are "in the labor force" if you are available for work and either working or actively seeking work. The unemployment rate is the proportion of the labor force (not of the entire population) that is unemployed. Here are estimates from the Current Population Survey for the civilian population aged 25 years and over in a recent year. The table entries are counts in thousands of people.

Unemployment Suppose that you randomly select one person 25 years of age or older.

a. What is the probability that a randomly chosen person 25 years of age or older is in the labor force?

b. If you know that a randomly chosen person 25 years of age or older is a college graduate, what is the probability that he or she is in the labor force?

c. Are the events "in the labor force" and "college graduate" independent? Justify your answer.

Songs on an iPod Refer to Exercise 53 . How many songs would you need to sample if you wanted the standard deviation of the sampling distribution ofx-x炉 to be 10 seconds? Justify your answer.

More sample proportions List all 4possible SRSs of size n=3, calculate the proportion of red cars in the sample, and display the sampling distribution of the sample proportion on a dot plot with the same scale as the dot plot in Exercise 19. How does the variability of this sampling distribution compare with the variability of the sampling distribution from Exercise 19? What does this indicate about increasing the sample size?

From exercise19:

Car NumberColorAge
1
Red
1
2
White
5
3
Silver
8
4
Red
20
See all solutions

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