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Chapter 8: Problem 55

The first day of baseball comes in late March, ending in October with the World Series. Does fan support grow as the season goes on? Two CNN/USA Today/Gallup polls, one conducted in March and one in November, both involved random samples of 1001 adults aged 18 and older. In the March sample, \(45 \%\) of the adults claimed to be fans of professional baseball, while \(51 \%\) of the adults in the November sample claimed to be fans. \({ }^{13}\) a. Construct a \(99 \%\) confidence interval for the difference in the proportion of adults who claim to be fans in March versus November. b. Does the data indicate that the proportion of adults who claim to be fans increases in November, around the time of the World Series? Explain.

Short Answer

Expert verified
Answer: Yes, we can conclude that the proportion of adults who claim to be fans of professional baseball is higher in November than in March, as the entire 99% confidence interval is negative, indicating that fan support grew as the season progressed through November.

Step by step solution

01

Calculate the point estimate of the difference in proportions

The point estimate of the difference in proportions is given by $$\hat{p}_1 - \hat{p}_2,$$ where \(\hat{p}_1\) and \(\hat{p}_2\) are the sample proportions. In this case, we have: - \(\hat{p}_1 = 0.45\) (proportion of adults who claimed to be fans in March) - \(\hat{p}_2 = 0.51\) (proportion of adults who claimed to be fans in November) So the point estimate for the difference in proportions is: $$\hat{p}_1 - \hat{p}_2 = 0.45 - 0.51 = -0.06$$
02

Calculate the standard error

The standard error for the difference in proportions is given by $$SE(\hat{p}_1 - \hat{p}_2) = \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}},$$ where \(n_1\) and \(n_2\) are the sample sizes. In this case, we have: - \(n_1 = 1001\) (number of adults in the March sample) - \(n_2 = 1001\) (number of adults in the November sample) So the standard error is: $$SE(\hat{p}_1 - \hat{p}_2) = \sqrt{\frac{0.45(1 - 0.45)}{1001} + \frac{0.51(1 - 0.51)}{1001}} \approx 0.022$$
03

Construct the confidence interval

Now, we can construct the \(99\%\) confidence interval for the difference in proportions. For a \(99\%\) confidence level, the critical value (z-score) is approximately \(2.576\). Therefore, the confidence interval is given by $$CI = (\hat{p}_1 - \hat{p}_2) \pm z \times SE(\hat{p}_1 - \hat{p}_2),$$ where - \(CI\) is the confidence interval - \(z\) is the critical value (z-score) Plugging the values, we get $$CI = -0.06 \pm 2.576 \times 0.022 \approx (-0.107, -0.013)$$
04

Interpret the confidence interval

Based on the \(99\%\) confidence interval, we can say that we are \(99\%\) confident that the true difference in the proportion of adults who claim to be fans in March versus November is between \(-10.7\%\) and \(-1.3\%\). Since the entire confidence interval is negative, we can conclude that the proportion of adults who claim to be fans of professional baseball is higher in November than in March. b. Yes, the data indicates an increase in the proportion of adults who claim to be fans around the time of the World Series in November. The difference in proportions (-10.7%, -1.3%) in the \(99\%\) confidence interval indicates that fan support grew as the season progressed through November. Since the entire confidence interval is negative, it suggests that the proportion of adults who claim to be fans is higher in November than in March.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference in Proportions
Understanding the difference in proportions is a fundamental concept in statistics, especially when comparing two different groups or time periods. In the context of our exercise, we are comparing the proportion of adults claiming to be baseball fans in two distinct months: March and November. The difference in proportions is represented mathematically as \( \hat{p}_1 - \hat{p}_2 \), where \( \hat{p}_1 \) is the sample proportion at one time point (March), and \( \hat{p}_2 \) is the sample proportion at another time point (November).

For the baseball fan example, the difference in proportions is \( -0.06 \) which indicates that the fan base in November is higher by 6 percentage points when compared to March. This seemingly simple comparison provides a basis for further analysis, such as determining if there is a statistically significant increase in fan support towards the end of the baseball season, around the time of the World Series.
Standard Error Calculation
The standard error (SE) of the difference in proportions measures the variability of the difference estimate due to sampling. It is crucial for constructing confidence intervals and hypothesis testing because it indicates how precise our estimate of the difference in proportions is.

To calculate the standard error in the case of two independent proportions, we use the formula:\[SE(\hat{p}_1 - \hat{p}_2) = \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}\]where \(n_1\) and \(n_2\) are the sample sizes for March and November, respectively. With a standard error of approximately \(0.022\), we can then apply this value to our confidence interval formula to determine the range in which the true difference in proportions is likely to be found.
Hypothesis Testing
Hypothesis testing is a systematic method used to make statistical decisions using experimental data. Hypothesis testing is based on concepts such as the null hypothesis, which proposes no effect or no difference, while the alternative hypothesis represents what we aim to prove or detect - an effect or a difference. In our problem, the null hypothesis might state that there is no difference in the proportion of baseball fans from March to November.

By constructing a confidence interval, conducting hypothesis testing allows us to assess whether the observed difference in proportions is statistically significant. The confidence interval from the exercise, \( (-0.107, -0.013) \), tells us that, at a 99% confidence level, the proportion of fans in November is significantly higher than in March. Since the interval does not contain zero, which would indicate no difference, we reject the null hypothesis and accept the alternative hypothesis, concluding there is a noticeable increase in the fan base during the World Series.

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Most popular questions from this chapter

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