Problem 17 - Basic Statistics for Business and Economics with Student

Question

The Fox TV network is considering replacing one of its prime-time crime investigation shows with a new familyoriented comedy show. Before a final decision is made, network executives commission a sample of 400 viewers. After viewing the comedy, 250 indicated they would watch the new show and suggested it replace the crime investigation show. a. Estimate the value of the population proportion. b. Develop a 99 percent confidence interval for the population proportion. c. Interpret your findings.

Step-by-Step Solution

Answer

The population proportion is 0.625; the 99% confidence interval is [0.5627, 0.6873].

Step 1: Estimate the Population Proportion

The given problem states that out of a sample of 400 viewers, 250 viewers indicated they would watch the new show. To estimate the population proportion \( p \), use the formula: \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successes and \( n \) is the sample size. Here, \( x = 250 \) and \( n = 400 \). Thus, \( \hat{p} = \frac{250}{400} = 0.625 \).

Step 2: Determine the Standard Error

Compute the standard error for the sample proportion, which will be used to develop the confidence interval. The formula for the standard error \( SE \) is: \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \). Substituting the values, we get \( SE = \sqrt{\frac{0.625 \times 0.375}{400}} \approx 0.0242 \).

Step 3: Find the Z-Score for a 99 Percent Confidence Interval

For a 99% confidence interval, the Z-score is typically 2.576 (you can find this in Z-tables which provide the critical values of the standard normal distribution).

Step 4: Compute the Confidence Interval

Use the formula for the confidence interval: \( \hat{p} \pm Z \times SE \). Substituting in the values we have: \( 0.625 \pm 2.576 \times 0.0242 \). This gives us an interval: \( 0.625 \pm 0.0623 \), which results in \( [0.5627, 0.6873] \).

Step 5: Interpretation of the Confidence Interval

The confidence interval \([0.5627, 0.6873]\) suggests that we are 99% confident that the true population proportion of viewers who would prefer the new comedy show lies between 56.27% and 68.73%.

Key Concepts

Population Proportion

The population proportion is a pivotal concept in statistics. It represents the fraction of a population that displays a particular characteristic. In this scenario, it relates to the proportion of TV viewers who favor the new comedy show over the existing crime investigation show. To calculate the estimated population proportion, we rely on the data collected from our sample. With 250 out of 400 viewers advocating for the comedy show, the estimator formula, \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of viewers in favor and \( n \) is the total sample size, is applied. Hence, \( \hat{p} = \frac{250}{400} = 0.625 \). This estimate suggests that around 62.5% of the viewer population may support the new comedy show. Remember, this is just an estimate based on the sample.

Sample Size

The concept of sample size, denoted as \( n \), is crucial in statistical analysis. In this context, the Fox TV network used a sample of 400 viewers. Sample size significantly impacts the precision of our estimates. A larger sample size generally leads to more reliable results. It minimizes the margin of error and improves the confidence of our population proportion estimate. In other words, if Fox TV could sample more than 400 viewers, they might achieve a narrower confidence interval, implying greater certainty about their estimate of the population's preference for the new show.

Standard Error

Standard error (SE) measures the variability or precision of the sample proportion estimate. It tells us how much the sample proportion might differ from the real population proportion. To calculate the standard error, we use the formula: \[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] Substituting in the sample proportion \( \hat{p} = 0.625 \) and sample size \( n = 400 \), we find: \[ SE = \sqrt{\frac{0.625 \times 0.375}{400}} \approx 0.0242 \] This value means that the sample proportion is estimated to fluctuate by about 2.42% if different viewers are sampled.A smaller SE indicates more precision in estimating the population proportion, reinforcing the importance of a larger sample size.

Z-Score

The Z-score is a statistical metric that describes how much a particular value deviates from the mean, measured in terms of standard deviations. In confidence interval calculations, it determines the level of certainty desired for the estimate. For a 99% confidence interval, the Z-score is typically 2.576. This indicates a very high confidence level, reflecting how far out in the tails of the standard normal distribution you reach. To develop a confidence interval incorporating the Z-score, you use the formula: \( \hat{p} \pm Z \times SE \)Applying this formula with the known Z-score and standard error allows us to determine the range where the actual population proportion may lie. In the example, this interval is \([0.5627, 0.6873]\), meaning we are 99% confident that the true proportion of viewers who prefer the new show will be between 56.27% and 68.73%.