A recent survey collected information on television viewing habits from …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

A recent survey collected information on television viewing habits from a random sample of 1,000 people in the United States. Of those sampled, 37 percent indicated that their favorite sport to watch on television was American football. Construct and interpret a 95 percent confidence interval for the proportion of all people in the United States who would indicate that their favorite sport to watch on television is American football. Based on your answer to the previous problem, is it reasonable to believe that 33 percent is the actual percent of people in the United States whose favorite sport to watch on television is American football

Short Answer

To calculate a 95% confidence interval for the proportion of Americans favoring American football, a sample of 1000 showed that 37% preferred the sport, resulting in a CI of (0.34, 0.40). Since the null hypothesis of 33% falls outside this range, it is rejected as a valid representation of the population’s preference.

Step-by-Step Solution

Step 1: Understand the Sample and Proportion

To establish a 95% Confidence Interval (CI) for the proportion of people in the United States who favor American football, we work with a sample size of 1000 individuals. In this sample, 37% have identified American football as their favorite sport to watch on television, which converts to a probability of p = 0.37.

Step 2: Calculate the Confidence Interval

Using the formula for the confidence interval, apply the following parameters: the sample proportion (p), sample size (n), and the z-score for the desired confidence level. The formula is: CI = p ¬± z‚CA‚ao(p(1-p)/n). For 95% confidence, the z-score (z‚CA) is 1.96. Plugging in the values, we compute:

  • CI = 0.37 ± 1.96 ‚Äöao(0.37 ‚àöo 0.63 / 1000)
  • This results in a CI of (0.34, 0.40).

Step 3: Evaluate the Null Hypothesis

To determine if 33% (or 0.33) can be accepted as a reasonable representation of viewers who prefer American football, we compare it to our computed confidence interval limits: 0.34 and 0.40. Since 0.33 lies below the lower limit of the CI (0.34), we reject the hypothesis that the proportion of people who like football is 33%. Hence, it is unreasonable to believe that 33% accurately represents this population.

Related Concepts

Sample Proportion

The ratio of individuals in a sample who have a specific characteristic, used to estimate the true proportion in the population

Confidence Interval (Ci)

A range of values derived from a sample that is likely to contain the true population parameter, expressed with a specified level of confidence (such as 95%)

Z-Score

A statistical measurement that describes a value’s relationship to the mean of a group of values, used in statistics for calculating confidence intervals.

Table Of Contents
  1. A recent survey collected information on television viewing habits from a random sample of 1,000 people in the United States. Of those sampled, 37 percent indicated that their favorite sport to watch on television was American football. Construct and interpret a 95 percent confidence interval for the proportion of all people in the United States who would indicate that their favorite sport to watch on television is American football. Based on your answer to the previous problem, is it reasonable to believe that 33 percent is the actual percent of people in the United States whose favorite sport to watch on television is American football

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