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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 μ=9.70 ounces and standard deviation σ=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 (μ)=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-μσ<9.65-μσ=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¯<9.65)=Px¯-μσn<9.65-μσn=PZ<9.65-9.700.035¯=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.

Sample minimums List all 10possible SRSs of size n=3, calculate the minimum quiz score for each sample, and display the sampling distribution of the sample minimum on a dotplot.

Making auto parts A grinding machine in an auto parts plant prepares axles with a target diameter μ=40.125millimeters (mm). The machine has some variability, so the standard deviation of the diameters is σ=0.002mm. The machine operator inspects a random sample of 4 axles each hour for quality control purposes and records the sample mean diameter x-x¯. Assume the machine is working properly.

a. Identify the mean of the sampling distribution of x-·x¯.

b. Calculate and interpret the standard deviation of the sampling distribution of x-x¯.

Lightning strikes The number of lightning strikes on a square kilometer of open ground in a year has mean 6 and standard deviation 2.4. The National Lightning Detection Network (NLDN) uses automatic sensors to watch for lightning in 1-square-kilometer plots of land. Find the probability that the total number of lightning strikes in a random sample of 50 square-kilometer plots of land is less than 250.

Suppose we select an SRS of size n=100n=100from a large population having proportion p of successes. Let p∧p^be the proportion of successes in the sample. For which value of p would it be safe to use the Normal approximation to the sampling distribution of p∧p^?

a. 0.01

b. 0.09

c. 0.85

d. 0.975

e. 0.999
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