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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 6 - Confidence IntervalsProblema 6.R.3
Capitolo 6, Problema 6.R.3

(a) Construct a 90% confidence interval for the population mean in Exercise 1. Interpret the results. (b) Does it seem likely that the population mean could be within 10% of the sample mean? Explain.

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Step 1: Identify the necessary components for constructing a confidence interval. You will need the sample mean (\( \bar{x} \)), the sample standard deviation (\( s \)), the sample size (\( n \)), and the critical value (\( t^* \)) for a 90% confidence level. The critical value can be found using a t-distribution table or statistical software, based on the degrees of freedom \( df = n - 1 \).
Step 2: Use the formula for the confidence interval for the population mean: \( \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}} \). Substitute the values for \( \bar{x} \), \( t^* \), \( s \), and \( n \) into the formula to calculate the lower and upper bounds of the confidence interval.
Step 3: Interpret the confidence interval. A 90% confidence interval means that if we were to take many random samples and construct confidence intervals for each, approximately 90% of those intervals would contain the true population mean. State the interval in the context of the problem.
Step 4: To determine if the population mean could be within 10% of the sample mean, calculate 10% of the sample mean (\( 0.1 \cdot \bar{x} \)) and check if this range falls entirely within the confidence interval. If it does, it is likely; if not, it is unlikely.
Step 5: Provide an explanation based on the results. If the range of 10% around the sample mean is within the confidence interval, explain why this suggests the population mean could plausibly be within that range. If not, explain why the confidence interval suggests otherwise.

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Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence, such as 90%. It is calculated using the sample mean, the standard error, and a critical value from the t-distribution or z-distribution, depending on the sample size and whether the population standard deviation is known.
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Introduction to Confidence Intervals

Population Mean

The population mean is the average of all possible values in a population. It is a parameter that represents the central tendency of the entire group, as opposed to the sample mean, which is calculated from a subset of the population. Understanding the difference between these two means is crucial for making inferences about the population based on sample data.
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Population Standard Deviation Known

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It indicates how much the sample mean is expected to differ from the true population mean. In the context of confidence intervals, a smaller margin of error suggests a more precise estimate, while a larger margin of error indicates greater uncertainty about the population mean's location relative to the sample mean.
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Finding the Minimum Sample Size Needed for a Confidence Interval
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In a random sample of 36 top-rated roller coasters, the average height is 165 feet and the standard deviation is 67 feet. Construct a 90% confidence interval for μ. Interpret the results. (Source: POP World Media, LLC)

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In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.

c = 0.98, s = 0.9, n = 12, xbar = 6.8

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In Exercises 5 and 6, use the confidence interval to find the margin of error and the sample mean.

(7.428, 7.562)

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Determine the minimum sample size required to be 99% confident that the sample mean driving distance to work is within 2 miles of the population mean driving distance to work. Use the population standard deviation from Exercise 2.

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In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.

c = 0.90, s = 25.6, n = 16, xbar = 72.1

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In Exercise 19, would it be unusual for the population proportion to be 38%? Explain.

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