7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

2:38 minutes

Problem 12.CR.6b

Textbook Question

Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).
b. If 25 quarters are randomly selected, find the probability that their mean weight is greater than 5.675 g.

Verified step by step guidance

Step 1: Identify the given values in the problem. The population mean (μ) is 5.670 g, the population standard deviation (σ) is 0.062 g, and the sample size (n) is 25. The problem asks for the probability that the sample mean weight is greater than 5.675 g.

Step 2: Calculate the standard error of the mean (SEM). The SEM is given by the formula:

σ / \sqrt{n}

, where σ is the population standard deviation and n is the sample size.

Step 3: Standardize the sample mean using the z-score formula:

z = (X - μ)/ SEM)

. Here, X is the sample mean (5.675 g), μ is the population mean (5.670 g), and SEM is the standard error of the mean calculated in Step 2.

Step 4: Use the z-score obtained in Step 3 to find the corresponding probability. This can be done by looking up the z-score in a standard normal distribution table or using statistical software to find the cumulative probability.

Step 5: Subtract the cumulative probability from 1 to find the probability that the sample mean weight is greater than 5.675 g. This is because the problem asks for the probability in the upper tail of the distribution.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this context, the weights of quarters follow a normal distribution, which allows us to use statistical methods to calculate probabilities related to their mean.

Central Limit Theorem

The Central Limit Theorem states that the sampling distribution of the sample mean will be normally distributed, regardless of the shape of the population distribution, provided the sample size is sufficiently large (typically n > 30). In this case, with a sample size of 25, we can still apply the theorem to approximate the distribution of the sample mean of quarter weights.

Z-Score

A Z-score measures how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the value and dividing by the standard deviation. In this problem, we will use the Z-score to determine the probability that the mean weight of the selected quarters exceeds 5.675 g by standardizing the sample mean.

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Textbook Question

Minting Quarters Listed below are weights (grams) of quarters minted after 1964 (based on Data Set 40 “Coin Weights” in Appendix B). Construct a 95% confidence interval estimate of the mean weight of all quarters minted after 1964. Specifications require that the quarters have a weight of 5.670 g. What does the confidence interval suggest about that specification?

Textbook Question

Los Angeles Commute Time Listed below are 15 Los Angeles commute times (based on a sample from Data Set 31 “Commute Times” in Appendix B). Construct a 99% confidence interval estimate of the population mean. Is the confidence interval a good estimate of the population mean?

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Genes Samples of DNA are collected, and the four DNA bases of A, G, C, and T are coded as 1, 2, 3, and 4, respectively. The results are listed below. Construct a 95% confidence interval estimate of the mean. What is the practical use of the confidence interval?

2 2 1 4 3 3 3 3 4 1

Textbook Question

Mercury in Sushi An FDA guideline is that the mercury in fish should be below 1 part per million (ppm). Listed below are the amounts of mercury (ppm) found in tuna sushi sampled at different stores in New York City. The study was sponsored by the New York Times, and the stores (in order) are D’Agostino, Eli’s Manhattan, Fairway, Food Emporium, Gourmet Garage, Grace’s Marketplace, and Whole Foods. Construct a 98% confidence interval estimate of the mean amount of mercury in the population. Does it appear that there is too much mercury in tuna sushi?

0.56 0.75 0.10 0.95 1.25 0.54 0.88

Textbook Question

Requirements A construction quality control analyst has collected a random sample of six concrete road barriers, and she plans to weigh each of them and construct a 95% confidence interval estimate of the mean weight of all such barriers. What requirements must be satisfied in order to construct the confidence interval with the method from Section 7-2 that uses the t distribution?

Textbook Question

Degrees of Freedom In general, what does “degrees of freedom” refer to? For the sample data described in Exercise 7 “Requirements,” find the number of degrees of freedom, assuming that you want to construct a confidence interval estimate of u using the t distribution.

Textbook Question

Space Mountain Use the following wait times (minutes) for the Space Mountain ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B). Construct a 95% confidence interval estimate of the mean of all wait times. Write a brief statement that interprets that confidence interval.

40 35 40 40 25 80 50 30 35 40

Textbook Question

In Exercises 1–4, refer to the accompanying screen display that results from a simple random sample of times (minutes) between eruptions of the Old Faithful geyser. The confidence level of 95% was used.

Refer to the accompanying screen display.

a. Express the confidence interval in the format that uses the “less than” symbol. Round the confidence interval limits given that the original times are all rounded to one decimal place.

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Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).b. If 25 quarters are randomly selected, find the probability that their mean weight is greater than 5.675 g.

Key Concepts

Normal Distribution

Central Limit Theorem

Z-Score

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Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).
b. If 25 quarters are randomly selected, find the probability that their mean weight is greater than 5.675 g.