Problem 1 - AP Statistics with 6 Practice Tests

Question

Researchers are investigating the association between humidity (percent) and attendance (1000s) at major league baseball games for a random sample of days and stadiums. A 95 percent confidence interval for the slope is constructed for predicting attendance based on humidity, obtaining (-1.32, o.18). Which statement is correct? (A) The association is probably negative because most of the interval is below o. (B) The association must be negative because the interval contains -1 (C) The association is not significant because o is in the interval. (D) The relatively high margin of error, \(0.75,\) makes any conclusion suspect. (E) A calculation error was made because the lower endpoint is below -1

Step-by-Step Solution

Answer

(C) The association is not significant because 0 is in the interval.

Step 1: Understanding the Confidence Interval

A 95 percent confidence interval for the slope in a linear regression context provides a range within which the true slope parameter lies, with 95% confidence. Here, the interval given is (-1.32, 0.18).

Step 2: Assessing the Interval's Implications

The interval includes negative and positive values. This means that there is uncertainty about the direction of the slope. If the interval were entirely below 0, the slope would be significantly negative. If entirely above 0, it would be significantly positive.

Step 3: Evaluating the Statements

Each statement must be evaluated based on the implications of the interval (-1.32, 0.18).- (A) The association is probably negative because most of the interval is below 0: This is incorrect as it is not about 'most' of the interval; rather, it matters if 0 is within the interval or not.- (B) The association must be negative because the interval contains -1: This is incorrect, the presence of -1 doesn’t guarantee negativity since the interval also includes positive values.- (C) The association is not significant because 0 is in the interval: This is correct. Since the confidence interval includes 0, the slope is not statistically significant at the 95% confidence level.- (D) The relatively high margin of error, 0.75, makes any conclusion suspect: The margin of error being high is not directly inferred from the interval bounds, hence less relevant.- (E) A calculation error was made because the lower endpoint is below -1: This is incorrect as the lower endpoint can reasonably be less than -1.

Step 4: Conclusion

Statement (C) is the correct one since the 95% confidence interval includes 0, hence the slope is not statistically significant.

Key Concepts

slope significance

The significance of the slope in a linear regression analysis helps us understand whether the relationship between the predictor and the response variable is statistically significant. In this case, the predictor is humidity, and the response variable is attendance at baseball games. The slope tells us the change in attendance for each unit change in humidity.
For example, a negative slope means that as humidity increases, attendance tends to decrease, and a positive slope means the opposite. To determine if this slope is statistically significant, we look at the confidence interval. If the confidence interval for the slope includes zero, it means there is no significant evidence to say that a relationship between the variables exists because it's possible the true slope could be zero. For the given interval (-1.32, 0.18), zero is included, suggesting that the association between humidity and attendance is not statistically significant.

interpretation of confidence intervals

A confidence interval provides a range of values within which we believe the true parameter lies, with a certain level of confidence. For a 95% confidence interval, we can say that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect about 95 of those intervals to contain the true parameter value.
In this exercise, we have a 95% confidence interval for the slope of the regression line that predicts baseball game attendance based on humidity, which is (-1.32, 0.18). Since this interval includes both negative and positive values, it indicates uncertainty over the direction of the association. The interval being mostly negative doesn’t ensure a negative association; what really matters is whether zero is included in the interval.
Here, since the interval includes zero, we are not confident that there is a significant relationship between humidity and attendance.

linear regression analysis

Linear regression analysis is a method to model the relationship between a dependent variable and one or more independent variables. By fitting a line through the data, we can make predictions and understand the strength and direction of the relationships.
In the exercise, researchers are studying how humidity affects attendance at baseball games. The linear regression model will provide a slope that describes how much attendance changes for each one-unit increase in humidity. The confidence interval around this slope tells researchers how precise their estimate is.
A narrower confidence interval indicates a more precise estimate of the slope, while a wider interval suggests greater uncertainty. Given the confidence interval (-1.32, 0.18), the wide range reflects higher variability and uncertainty in predicting attendance from humidity alone.

association in datasets

Association in datasets refers to the relationship between two or more variables. This can be positive (as one variable increases, the other also increases), negative (as one variable increases, the other decreases), or non-significant (no clear pattern).
In the context of the given exercise, the researchers are exploring the association between humidity and baseball game attendance. By constructing a confidence interval for the slope, they aim to understand if there is a statistically significant relationship. However, since the interval (-1.32, 0.18) includes zero, it suggests no significant association between humidity and attendance.
Finding an association (or lack thereof) can have practical implications, such as guiding decisions on scheduling games to maximize attendance or understanding environmental impacts on spectator sports.