Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.T.3c

Chapter 6, Problem 6.T.3c

The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)

c. Would it be unusual for the population mean to be under 575? Explain.

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Step 1: Calculate the sample mean (x̄) using the provided data set. Add all the scores together and divide by the total number of scores (12). The formula is x̄ = (Σx) / n, where Σx is the sum of all scores and n is the number of scores.

Step 2: Determine the standard error of the mean (SE). The formula for SE is SE = σ / √n, where σ is the population standard deviation (108) and n is the sample size (12).

Step 3: Calculate the z-score to determine how far the sample mean is from the hypothesized population mean of 575. The formula for the z-score is z = (x̄ - μ) / SE, where μ is the hypothesized population mean (575).

Step 4: Use the z-score to find the corresponding probability (p-value) from the standard normal distribution table. This will indicate the likelihood of observing a sample mean as extreme as the calculated mean if the population mean were truly 575.

Step 5: Interpret the p-value. If the p-value is less than 0.05 (or another chosen significance level), it would be unusual for the population mean to be under 575. Otherwise, it would not be considered unusual.

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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 SAT Physics scores are assumed to follow a normal distribution, which allows for the application of statistical methods to analyze the data and make inferences about the population.

Population Mean

The population mean is the average of all possible values in a population. It is a key parameter in statistics, as it provides a measure of central tendency. In this question, determining whether the population mean could be under 575 involves comparing it to the expected distribution of scores based on the provided data and standard deviation.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In this case, the population standard deviation of 108 helps assess how unusual it would be for the population mean to fall below a certain threshold, such as 575.

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Textbook Question

The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)

b. Construct a 90% confidence interval for the population mean. Interpret the results.

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Textbook Question

Use the standard normal distribution or the t-distribution to construct the indicated confidence interval for the population mean of each data set. Justify your decision. If neither distribution can be used, explain why. Interpret the results.

a. In a random sample of 40 patients, the mean waiting time at a dentist’s office was 20 minutes and the standard deviation was 7.5 minutes. Construct a 95% confidence interval for the population mean.

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Textbook Question

The data set represents the weights (in pounds) of 10 randomly selected black bears from northeast Pennsylvania. Assume the weights are normally distributed. (Source: Pennsylvania Game Commission)

b. Construct a 95% confidence interval for the population mean. Interpret the results.

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Textbook Question

d. Determine the minimum sample size required to be 95% confident that the sample mean test score is within 10 points of the population mean test score.

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Textbook Question

b. In a random sample of 15 cereal boxes, the mean weight was 11.89 ounces. Assume the weights of the cereal boxes are normally distributed and the population standard deviation is 0.05 ounce. Construct a 90% confidence interval for the population mean.

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Textbook Question

In a survey of 2096 U.S. adults, 1740 think football teams of all levels should require players who suffer a head injury to take a set amount of time off from playing to recover. (Adapted from The Harris Poll)

b. Construct a 95% confidence interval for the population proportion. Interpret the results.

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