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Chapter 7: Q. T7.4 (page 486)

The central limit theorem is important in statistics because it allows us to use a Normal distribution to find probabilities involving the sample mean if the

a. sample size is reasonably large (for any population).

b. population is Normally distributed (for any sample size).

c. population is Normally distributed and the sample size is reasonably large.

d. population is Normally distributed and the population standard deviation is known (for any sample size).

e. population size is reasonably large (whether the population distribution is known or not).

Short Answer

Expert verified

(a) The central limit theorem is important in statistics because it allows us to use a Normal distribution to find probabilities involving the sample mean if the sample size is reasonably large (for any population).

Step by step solution

01

Given Information

The central limit theorem is significant in statistics because it allows us to find probabilities involving the sample mean using a Normal distribution.

02

Explanation for correct option

According to the central limit theorem, if the sample size is big, the sampling distribution of the sample mean x¯is approximately normal.

As a result, the best solution is (a)

03

 Step 3: Explanation for incorrect option

b. population is Normally distributed (for any sample size) is not the answer.

c. population is Normally distributed and the sample size is reasonably large is not the answer.

d. population is Normally distributed and the population standard deviation is known (for any sample size) is not the answer.

e. population size is reasonably large (whether the population distribution is known or not) is not the answer.

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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’s 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?

Five books An author has written 5 children's books. The numbers of pages in these books are 64,66,71,73, and 76 .

a. List all 10 possible SRSs of size n=3,n=3, calculate the median number of pages for each sample, and display the sampling distribution of the sample median on a dotplot.

b. Describe how the variability of the sampling distribution of the sample median would change if the sample size was increased to n=4.n=4.

c. Construct the sampling distribution of the sample median for samples of size n=4. n=4. Does this sampling distribution support your answer to part (b)? Explain your reasoning.

Suppose that you are a student aide in the library and agree to be paid according to the "random pay" system. Each week, the librarian flips a coin. If the coin comes up heads, your pay for the week is 80\(. If it comes up tails, your pay for the week is 40\). You work for the library for 100 weeks. Suppose we choose an SRS of 2 weeks and calculate your average earnings x-x¯. The shape of the sampling distribution of will be

a. Normal.

b. approximately Normal.

c. right-skewed.

d. left-skewed.

e. symmetric but not Normal.

Squirrels and their food supply (3.2) Animal species produce more offspring when their supply of food goes up. Some animals appear able to anticipate unusual food abundance. Red squirrels eat seeds from pinecones, a food source that sometimes has very large crops. Researchers collected data on an index of the abundance of pinecones and the average number of offspring per female over 16years.4Computer output from a least-squares 4regression on these data and a residual plot are shown here

a. Is a linear model appropriate for these data? Explain.

b. Give the equation for the least-squares regression line. Define any variables you use.

c. Interpret the values of r2and sin context.

Detecting gypsy moths The gypsy moth is a serious threat to oak and aspen trees. A state agriculture department places traps throughout the state to detect the moths. Each month, an SRS of 50 traps is inspected, the number of moths in each trap is recorded, and the mean number of moths is calculated. Based on years of data, the distribution of moth counts is discrete and strongly skewed with a mean of 0.5 and a standard deviation of 0.7.

a. Explain why it is reasonable to use a Normal distribution to approximate the sampling distribution of x-x¯for SRSs of size 50 .

b. Estimate the probability that the mean number of moths in a sample of size 50 is greater than or equal to 0.6.

c. In a recent month, the mean number of moths in an SRS of size 50 was x-=0.6. x¯=0.6. Based on this result, is there convincing evidence that the moth population is getting larger in this state? Explain your reasoning.
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