Problem 53 - Basic Statistics for Business & Economics

Question

Based on a sample of 50 U.S. citizens, the American Film Institute found that a typical American spent 78 hours watching movies last year. The standard deviation of this sample was 9 hours. a. Develop a \(95 \%\) confidence interval for the population mean number of hours spent watching movies last year. b. How large a sample should be used to be \(90 \%\) confident the sample mean is within 1\. O hour of the population mean?

Step-by-Step Solution

Answer

a. [75.51, 80.49]; b. 220 samples.

Step 1: Identify the Required Data

To develop a 95% confidence interval for the population mean, we have the following data: the sample mean \(\bar{x} = 78\), the sample standard deviation \(s = 9\), and the sample size \(n = 50\).

Step 2: Determine the Z-score for a 95% Confidence Interval

For a 95% confidence level, the Z-score (or critical value) is approximately 1.96. This value is taken from the standard normal distribution table.

Step 3: Calculate the Standard Error of the Mean

The standard error (SE) is calculated using the formula: \[ SE = \frac{s}{\sqrt{n}} \] where \(s = 9\) is the sample standard deviation, and \(n = 50\) is the sample size. Thus, \[ SE = \frac{9}{\sqrt{50}} \approx 1.27. \]

Step 4: Develop the Confidence Interval

The 95% confidence interval is calculated using the formula: \[ \bar{x} \pm Z \times SE \] where \(\bar{x} = 78\), \(Z = 1.96\), and \(SE \approx 1.27\). Thus, \[ 78 \pm 1.96 \times 1.27 \approx 78 \pm 2.49. \] So the confidence interval is \([75.51, 80.49]\).

Step 5: Identify Required Data for Sample Size Calculation

For part b, determine the required sample size to be 90% confident that the sample mean is within 1 hour of the population mean. Known values: \(s = 9\), and the desired margin of error \(E = 1\).

Step 6: Determine the Z-score for a 90% Confidence Level

For a 90% confidence level, the Z-score is approximately 1.645.

Step 7: Calculate the Required Sample Size

The formula to calculate the required sample size \(n\) is: \[ n = \left(\frac{Z \times s}{E}\right)^2 \] Using \(Z = 1.645\), \(s = 9\), and \(E = 1\): \[ n = \left(\frac{1.645 \times 9}{1}\right)^2 = (14.805)^2 \approx 219.12. \] Since sample size must be a whole number, round up to 220.

Key Concepts

Sample Mean

When analyzing data from samples, the **sample mean** is a crucial statistic. It is simply the average of all the data points collected in your sample. In the given exercise, this is the average number of hours U.S. citizens spent watching movies, which is 78 hours per year. To compute the sample mean (\(\bar{x}\)), use the formula:- \[\bar{x} = \frac{\sum{x_i}}{n}\]- where \(x_i\) signifies each data point, and \(n\) is the total number of data points in the sample.The sample mean is used to make inferences about the population mean, aiding in the determination of parameters like the confidence interval. It's a powerful tool that helps summarize a large set of data into a single representative value.

Standard Deviation

The **standard deviation** quantifies the amount of variation or dispersion in a set of data. In the given example, the standard deviation of 9 hours indicates how much individual movie-watching hours differ from the average 78 hours.The formula to calculate standard deviation (\(s\)) is as follows:- \[s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\]- where \(x_i\) represents each data point, \(\bar{x}\) is the sample mean, and \(n\) is the sample size.A smaller standard deviation implies that data points are close to the mean, while a larger value indicates more spread out data.

Standard Error

The **standard error (SE)** measures how much the sample mean is expected to vary from the true population mean. It provides insight into the precision of the sample mean.To calculate the standard error, use the formula:- \[SE = \frac{s}{\sqrt{n}}\]- where \(s\) is the sample standard deviation, and \(n\) is the sample size.In our exercise, the standard error is approximately 1.27, calculated using a sample size of 50 and a standard deviation of 9. The smaller the standard error, the more reliable the sample mean as an estimate of the population mean.

Z-score

The **Z-score** is a statistical measure that expresses the number of standard deviations a data point is from the mean. It's pivotal in determining confidence intervals. For confidence interval calculations, a critical Z-score value is used. The exercise mentions Z-scores of 1.96 and 1.645, corresponding to 95% and 90% confidence levels, respectively. - At a 95% confidence level, the Z-score of 1.96 means that the interval captures the true population mean about 95% of the time. - Similarly, a 90% confidence level has a Z-score of 1.645. These values allow us to calculate a range wherein the true population parameter likely falls with a certain level of confidence.

Sample Size Calculation

Calculating the needed **sample size** is crucial for determining how many observations are required to make reliable population estimates. It ensures sufficient precision by controlling the margin of error.For the specified margin of error in the exercise, use the formula:- \[n = \left(\frac{Z \times s}{E}\right)^2\]- where \(Z\) is the Z-score for your chosen confidence level, \(s\) is the sample standard deviation, and \(E\) is the desired margin of error.In the example, the calculation determines that a sample size of 220 is required for 90% confidence, ensuring that the sample mean is within 1 hour of the true population mean. Rounding up the calculated value ensures the sample size is realistic and effectively captures the needed data characteristics.