Problem 57 - Basic Statistics for Business and Economics with Student

Question

A film alliance used a random sample of \(50 \mathrm{U.S.}\) citizens to estimate that the typical American spent 78 hours watching videos and DVDs last year. The standard deviation of this sample was 9 hours. a. Develop a 95 percent confidence interval for the population mean number of hours spent watching videos and DVDs last year. b. How large a sample should be used to be 90 percent confident the sample mean is within 1.0 hour of the population mean?

Step-by-Step Solution

Answer

a) 95% CI: (75.51, 80.49) hours. b) Sample size needed: 219.

Step 1: Understanding the Problem

We need to find a confidence interval for the mean and determine the sample size necessary for a specified confidence level and margin of error. Part (a) asks for a 95% confidence interval for the average time spent watching videos, while Part (b) requires finding the sample size for a 90% confidence interval within a 1-hour margin.

Step 2: Calculate the 95% Confidence Interval for Part (a)

Given the sample mean \(\bar{x} = 78\) hours, standard deviation \(s = 9\) hours, and sample size \(n = 50\). The Z-score for a 95% confidence level is approximately 1.96. We calculate the standard error: \(SE = \frac{s}{\sqrt{n}} = \frac{9}{\sqrt{50}}\). The confidence interval is:\[\bar{x} \pm Z \cdot SE = 78 \pm 1.96 \cdot \frac{9}{\sqrt{50}}\].

Step 3: Compute the Interval Values

Calculate the standard error: \(SE = \frac{9}{\sqrt{50}} \approx 1.2728\). The interval becomes:\[78 \pm 1.96 \times 1.2728 = (75.51, 80.49)\].Therefore, the 95% confidence interval is approximately between 75.51 and 80.49 hours.

Step 4: Determine Sample Size for Part (b)

To find the required sample size \(n\) for a 90% confidence interval within 1 hour, we use the formula:\[n = \left(\frac{Z \cdot s}{E}\right)^2\], where \(E = 1\) (margin of error) and \(Z \approx 1.645\) for a 90% confidence level. Substitute the values:\[n = \left(\frac{1.645 \times 9}{1}\right)^2 = (14.805)^2\].

Step 5: Round the Sample Size

Calculate \(n\) and round up to the nearest whole number since sample size must be an integer.\[n \approx 218.96 \rightarrow 219\]. Therefore, a sample size of 219 is necessary.

Key Concepts

Sample Size Determination

Determining the appropriate sample size is crucial when conducting research, especially if you want reliable results. A sample size that is too small may not represent the entire population, while a very large sample might be unnecessary and costly.
When aiming for a desired accuracy in results, especially in forming confidence intervals, the sample size plays a major role. In our case, to find out how many people we should survey to be 90% confident that our sample mean is within 1 hour of the true population mean, we use a simple formula. This formula looks like this:

• First, identify the Z-score, which is determined by your confidence level (for 90%, it is about 1.645).
• Next, multiply the Z-score by the standard deviation, then divide by the desired margin of error (1 hour in our case).
• Square this number to get your sample size: \( n = \left( \frac{Z \cdot s}{E} \right)^2 \).

Finally, it's important to round up, since a definite integer is needed as the sample size. For example, a calculated size of 218.96 would still require 219 respondents to ensure accuracy.

Standard Deviation

Standard deviation is a key concept in statistics that describes the spread or dispersion of a set of data. In other words, it indicates how much the individual data points deviate from the mean.
For the given exercise, the standard deviation was 9 hours, meaning that, on average, the time each sampled person watched videos deviated by around 9 hours from the mean of 78 hours. The formula for standard deviation is:

• Sum up the squared differences between each data point and the mean.
• Divide this sum by the sample size minus one (to get what's called the sample variance).
• Take the square root of the variance to have the standard deviation.

Standard deviation is pivotal in calculating other statistical values, such as the standard error used in confidence intervals. A smaller standard deviation suggests that data points are clustered closely around the mean, whereas a larger one suggests more variation.

Population Mean

The population mean is a central concept when estimating how much the entire population would average in a specific metric, using a sample. It is often based on sample data due to the impracticality of surveying an entire population.
In our case study, the mean number of hours per year U.S. citizens watch videos was estimated at 78 hours. This is termed the sample mean, but it acts as our best estimate of the population mean. The notation for this is commonly \( \bar{x} \) for the sample mean and \( \mu \) for the population mean.
Confidence intervals, like the one calculated in the exercise (75.51 to 80.49 hours), give a range that likely contains the true population mean. This calculation helps us understand the degree of certainty or uncertainty surrounding our estimate. In such analysis, confidence in our sample informs how closely our sample mean approximates the true population mean.