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Chapter 7: Q. 58 (page 479)

Cereal A company's cereal boxes advertise that each box contains 9.65 ounces of cereal. In fact, the amount of cereal in a randomly selected box follows a Normal distribution with mean 渭=9.70 ounces and standard deviation 蟽=0.03 ounce.

a. What is the probability that a randomly selected box of the cereal contains less than 9.65 ounces of cereal?

b. Now take an SRS of 5 boxes. What is the probability that the mean amount of cereal in these boxes is less than 9.65 ounces?

Short Answer

Expert verified
  1. The resultant probability is 0.0475.
  2. The resultant probability is 0.0001.

Step by step solution

01

Part (a) Step 1: Given information

Given:

The value of Population mean (渭)=9.70

The value of Population standard deviation(s)=0.03

02

Part (a) Step 2: Calculation

Consider the random variable X, which represents the amount of cereal in a box.

The likelihood that a randomly chosen box contains less than 9.65 ounces of cereal can be computed as follows:

P(X<9.65)=Px-渭蟽<9.65-渭蟽=PZ<9.65-9.700.03=P(Z<-1.67)=0.0475

Thus, the required probability is 0.0475.

03

Part (b) Step 1: Given information

The Sample size (n)=5

04

Part (b) Step 2: Calculation

The likelihood that the average amount of cereal in five randomly chosen boxes is less than 9.65ounces can be computed as follows:

P(X炉<9.65)=Px炉-渭蟽n<9.65-渭蟽n=PZ<9.65-9.700.035炉=P(Z<-3.73)=0.0001

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Most popular questions from this chapter

At a particular college, 78%of all students are receiving some kind of financial aid. The school newspaper selects a random sample of 100students and 72%of the respondents say they are receiving some sort of financial aid. Which of the following is true?

  1. 78%is a population and 72%is a sample.
  2. 72%is a population and 78%is a sample.
  3. 78%is a parameter and 72%is a statistic.
  4. 72%is a parameter and 78%is a statistic.
  5. 72%is a parameter and 100is a statistic.

Dem bones (2.2) Osteoporosis is a condition in which the bones become brittle due to the loss of minerals. To diagnose osteoporosis, an elaborate apparatus measures bone mineral density (BMD). BMD is usually reported in a standardized form. The standardization is based on a population of healthy young adults. The World Health Organization (WHO) criterion for osteoporosis is a BMD score that is 2.5standard deviations below the mean for young adults. BMD measurements in a population of people similar in age and gender roughly follow a Normal distribution.

a. What percent of healthy young adults have osteoporosis by the WHO criterion?

b. Women aged 70to 79are, of course, not young adults. The mean BMD in this age group is about-2 on the standard scale for young adults. Suppose that the standard deviation is the same as for young adults. What percent of this older population has osteoporosis?

More sample minimums List all 4possible SRSs of size n=3, calculate the minimum age for each sample, and display the sampling distribution of the sample minimum on a dot plot with the same scale as the dot plot in Exercise 20. How does the variability of this sampling distribution compare with the variability of the sampling distribution from Exercise 20? What does this indicate about increasing the sample size?

From exercise20:

Car NumberColorAge
1
Red1
2
White5
3
Silver8
4
Red20

A statistic is an unbiased estimator of a parameter when

a. the statistic is calculated from a random sample.

b. in a single sample, the value of the statistic is equal to the value of the parameter.

c. in many samples, the values of the statistic are very close to the value of the

parameter.

d. in many samples, the values of the statistic are centered at the value of the parameter.

e. in many samples, the distribution of the statistic has a shape that is approximately

Normal.

More homework Some skeptical Ap庐 Statistics students want to investigate the newspaper's claim in Exercise 11, so they choose an SRS of 100students from the school to interview. In their sample, 45students completed their homework last week. Does this provide convincing evidence that less than 60%of all students at the school completed their assigned homework last week?

a. What is the evidence that less than 60%of all students completed their assigned homework last week?

b. Provide two explanations for the evidence described in part (a).

We used technology to simulate choosing 250SRSs of size n=100n=100from a population of 2000students where 60%completed their assigned homework last week. The dotplot shows pp^the sample proportion of students who completed their assigned homework last week for each of the 250simulated samples.

c. There is one dot on the graph at 0.73. Explain what this value represents.

d. Would it be surprising to get a sample proportion of p=0.45p^=0.45or smaller in an SRS of size 100when p=0.60p=0.60? Justify your answer.

e. Based on your previous answers, is there convincing evidence that less than 60%of all students at the school completed their assigned homework last week? Explain your reasoning.
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