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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 6.T.3b
Chapter 6, Problem 6.T.3b

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)
Data set of 12 SAT Physics Subject Test scores: 590, 650, 730, 560, 460, 400, 620, 780, 510, 700, 590, 670.
b. Construct a 90% confidence interval for the population mean. Interpret the results.

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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 number of scores (12). The formula is x̄ = (Σx) / n.
Step 2: Identify the population standard deviation (σ), which is given as 108, and the sample size (n), which is 12.
Step 3: Determine the z-value corresponding to a 90% confidence level. For a 90% confidence interval, the z-value is approximately 1.645 (from the standard normal distribution table).
Step 4: Calculate the margin of error (E) using the formula E = z * (σ / √n). Substitute the values for z, σ, and n into the formula.
Step 5: Construct the confidence interval using the formula: Confidence Interval = x̄ ± E. Interpret the results by explaining that the interval provides a range within which the true population mean is likely to fall with 90% confidence.

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Key Concepts

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

Confidence Interval

A confidence interval is a range of values, derived from a data set, that is likely to contain the population parameter with a specified level of confidence. For example, a 90% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 90% of those intervals would contain the true population mean.
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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 assumption that the population test scores are normally distributed allows for the use of specific statistical methods, such as constructing confidence intervals.
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Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this case, the population standard deviation of 108 indicates how much the SAT Physics Subject Test scores deviate from the mean score. It is crucial for calculating the margin of error when constructing the confidence interval.
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Related Practice
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)

c. Construct a 99% confidence interval for the population standard deviation. Interpret the results.

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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)

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

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.

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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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)

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

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