Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

Problema 7.RE.4

Capitolo 7, Problema 7.RE.4

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

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Step 1: Calculate the sample mean (\( \bar{x} \)) of the given wait times. Add all the wait times together and divide by the total number of observations. The formula is \( \bar{x} = \frac{\sum x_i}{n} \), where \( x_i \) represents each wait time and \( n \) is the number of observations.

Step 2: Compute the sample standard deviation (\( s \)) using the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \). This measures the spread of the wait times around the mean.

Step 3: Determine the standard error of the mean (\( SE \)) using the formula \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the sample standard deviation and \( n \) is the sample size.

Step 4: Identify the critical value (\( t \)) for a 95% confidence interval using a t-distribution table. The degrees of freedom (\( df \)) are \( n-1 \). For a 95% confidence level, find the corresponding \( t \)-value based on \( df \).

Step 5: Construct the confidence interval using the formula \( \text{Confidence Interval} = \bar{x} \pm t \cdot SE \). This provides the range within which the true mean wait time is likely to fall. Interpret the interval by stating that we are 95% confident the true mean wait time lies within this range.

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Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically 95%. It provides an estimate of uncertainty around the sample mean, indicating how much the sample mean might vary from the actual population mean.

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06:33

Introduction to Confidence Intervals

Sample Mean

The sample mean is the average of a set of observations or data points. It is calculated by summing all the values in the sample and dividing by the number of observations. The sample mean serves as a point estimate of the population mean and is crucial for constructing confidence intervals.

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05:11

Sampling Distribution of Sample Proportion

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. It is essential for calculating the confidence interval, as it helps determine the margin of error.

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08:45

Calculating Standard Deviation

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b. Are the requirements for constructing a 95% confidence interval estimate of the population standard deviation satisfied? If so, construct that confidence interval.

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a. Among the 1002 people surveyed, what is the actual number of people who said that they voted?

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