Tennis Ball Manufacturing: A company manufactures tennis balls. When its tennis balls are dropped onto a concrete surface from a height of 100 inches, the company wants the mean height of a ball’s bounce upward to be 55.5 inches. This average is maintained by periodically testing random samples of 25 tennis balls. If a 99% confidence level contains the desired bounce height (55.5 inches), the company will be satisfied that it is manufacturing acceptable tennis balls. A sample of 25 tennis balls is randomly selected and tested. The mean bounce height of the sample is 56 inches and the standard deviation is .25 inches. Is the company making acceptable tennis balls? Explain your reasoning.
I have listed the givens, but I'm confused as to where the book is getting the t value of 2.797. I also don't know what equation to use
To determine whether the company is manufacturing acceptable tennis balls, we need to perform a hypothesis test and calculate a confidence interval.
Let's begin by setting up our null and alternative hypotheses:
Null Hypothesis (H0): The mean bounce height of tennis balls is equal to 55.5 inches.
Alternative Hypothesis (H1): The mean bounce height of tennis balls is not equal to 55.5 inches.
To perform the hypothesis test, we can use a t-test since we have a small sample size (n = 25) and do not know the population standard deviation. The t-value you mentioned, 2.797, appears to be a result of using the 99% confidence level and a two-tailed test.
The formula to calculate the t-value in this case is:
t = (sample mean - population mean) / (sample standard deviation / √sample size)
Let's plug in the given values:
sample mean (x̄) = 56 inches
population mean (μ) = 55.5 inches
sample standard deviation (s) = 0.25 inches
sample size (n) = 25
t = (56 - 55.5) / (0.25 / √25) = 2.797
Now, we need to compare the calculated t-value with the critical t-value to make our decision. The critical t-value can be found using the t-distribution table or statistical software. Since we have a two-tailed test, we want to consider both ends of the distribution.
At a 99% confidence level, the critical t-value for a two-tailed test with degrees of freedom (df) = n - 1 = 24 is approximately ±2.797.
Since our calculated t-value (2.797) falls within the range of the critical t-values (±2.797), we fail to reject the null hypothesis. This means that there is no significant evidence to suggest that the mean bounce height differs from the desired value of 55.5 inches.
Therefore, based on the given data, we can conclude that the company is making acceptable tennis balls.
Remember, hypothesis tests involve assumptions and limitations, so it's important to consider additional factors and consult statistical guidelines for your specific situation.
To determine whether the company is making acceptable tennis balls, we can conduct a hypothesis test using the given data. Here's how we can approach this problem step-by-step:
Step 1: State the null and alternative hypotheses.
- Null hypothesis (H0): The mean bounce height of the tennis balls is equal to the desired height of 55.5 inches.
- Alternative hypothesis (Ha): The mean bounce height of the tennis balls is not equal to 55.5 inches.
Step 2: Determine the significance level.
- The given confidence level is 99%. Therefore, the significance level (α) is calculated as 1 - confidence level = 1 - 0.99 = 0.01.
Step 3: Calculate the test statistic.
- To perform a hypothesis test, we can use the formula for the t-test:
t = (sample mean - population mean) / (sample standard deviation / √sample size)
- Given that the sample mean is 56 inches, the population mean (desired height) is 55.5 inches, the sample standard deviation is 0.25 inches, and the sample size is 25, we can calculate the t-value as follows:
t = (56 - 55.5) / (0.25 / √25) = 2.00
Step 4: Determine the critical value.
- Since the alternative hypothesis is two-tailed (the mean bounce height could be higher or lower than 55.5 inches), we need to find the critical t-value for a 99% confidence level with a significance level of 0.01/2 = 0.005.
- By referring to the t-distribution table or using statistical software, we find that the critical t-value is 2.797 for a two-tailed test with 24 degrees of freedom (sample size - 1).
Step 5: Compare the test statistic and critical value.
- Since the test statistic (t = 2.00) does not exceed the critical value (2.797), we fail to reject the null hypothesis.
Step 6: Draw a conclusion.
- Since we fail to reject the null hypothesis, we do not have sufficient evidence to conclude that the mean bounce height of the tennis balls is different from the desired height of 55.5 inches.
- This means that the company's tennis balls can be considered acceptable based on the 99% confidence level.
Therefore, according to the given data and the hypothesis test, the company is making acceptable tennis balls.