91Ó°ÊÓ

Injuries to Major League Baseball players have been increasing in recent years. For the period 1992 to 2001 , league expansion caused Major League Baseball rosters to increase \(15 \%\) However, the number of players being put on the disabled list due to injury increased \(32 \%\) over the same period (USA Today, July 8,2002 ). A research question addressed whether Major League Baseball players being put on the disabled list are on the list longer in 2001 than players put on the disabled list a decade earlier.a. Using the population mean number of days a player is on the disabled list, formulate null and alternative hypotheses that can be used to test the research question. b. Assume that the following data apply: \(\begin{array}{lll} & \mathbf{2 0 0 1} \text { Season } & \mathbf{1 9 9 2} \text { Season } \\ \text { Sample size } & n_{1}=45 & n_{2}=38 \\ \text { Sample mean } & \bar{x}_{1}=60 \text { days } & \bar{x}_{2}=51 \text { days } \\\ \text { Sample standard deviation } & s_{1}=18 \text { days } & s_{2}=15 \text { days }\end{array}\) What is the point estimate of the difference between population mean number of days on the disabled list for 2001 compared to \(1992 ?\) What is the percentage increase in the number of days on the disabled list? c. Use \(\alpha=.01 .\) What is your conclusion about the number of days on the disabled list? What is the \(p\) -value? d. Do these data suggest that Major League Baseball should be concerned about the situation?

Step by step solution

01

Define the Hypotheses

For part (a), we need to define the null and alternative hypotheses based on the given research question. The null hypothesis ( H_0) states there is no difference or decrease in days on the disabled list in 2001 compared to 1992. The alternative hypothesis ( H_a) posits that the days on the disabled list in 2001 are longer than in 1992. Thus, we define: - H_0: mu_1 leq mu_2 - H_a: mu_1 > mu_2 where mu_1 is the mean number of days on the disabled list in 2001, and mu_2 is the mean number of days in 1992.

02

Calculate the Point Estimate

For part (b), the point estimate of the difference in population means is the difference in sample means. Therefore, calculate the difference:\[\bar{x}_1 - \bar{x}_2 = 60 - 51 = 9 \\] Thus, the point estimate for the difference in mean number of days on the list between 2001 and 1992 is 9 days.

03

Calculate the Percentage Increase

Still within part (b), we want the percentage increase from 1992 to 2001. Use the formula:\[\text{Percentage Increase} = \left(\frac{\bar{x}_1 - \bar{x}_2}{\bar{x}_2}\right) \times 100 = \left(\frac{9}{51}\right) \times 100 \approx 17.65\%\]So, the percentage increase is approximately 17.65%.

04

Perform the Hypothesis Test

For part (c), use a two-sample t-test to test the hypotheses. The test statistic for comparing two means is:\[t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]Substitute the values:\[t = \frac{9}{\sqrt{\frac{18^2}{45} + \frac{15^2}{38}}} = \frac{9}{\sqrt{7.2 + 5.92}} = \frac{9}{3.467} \approx 2.595\]Now compare the test statistic to the critical value at \(\alpha = 0.01\) with degrees of freedom calculated approximately (using a software if needed) as 80.The critical value for H_a: mu_1 > mu_2 (one-tailed) at 0.01 is around 2.639, which is greater than 2.595, thus we do not reject H_0.

05

Determine the p-value and Conclusion

Using a t-distribution table or software, the p-value for \(t = 2.595\) with approximate DF 80 is around 0.01. Since the p-value is equal to our alpha level, it is right at the border, but conventionally, we cannot reject H_0 based on this test.

06

Interpret Implications

For part (d), while statistical significance wasn't strictly found at our alpha, it is close, suggesting players might indeed be on the list longer in 2001. This may warrant scrutiny and action to investigate causes for this increase in disabled list time.

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Key Concepts

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

t-test

A t-test is a statistical test that is used to determine if there is a significant difference between the means of two groups. In the context of Major League Baseball players on the disabled list, we use a two-sample t-test to decide if the mean number of days players spent on the list in 2001 was significantly longer than those in 1992. The calculation involves comparing the difference in means relative to the variability of the two samples.

• We compute the t-test statistic using the formula: \[ t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]
• Where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(s_1\) and \(s_2\) are the standard deviations, and \(n_1\) and \(n_2\) are the sample sizes of 2001 and 1992, respectively.
• The calculated t-value can then be compared to a critical value to test the hypothesis.

This method helps in verifying if observed differences can be attributed to random variation or if they reflect genuine differences.

null hypothesis

The null hypothesis (\(H_0\)) is a statement in hypothesis testing that assumes no effect or no difference. It serves as a starting point for statistical testing. When evaluating whether the length of time Major League Baseball players spend on the disabled list has changed from 1992 to 2001, the null hypothesis would claim there is no increase in time.

• Mathematically, this can be expressed as \(H_0: \mu_1 \leq \mu_2\).
• Here, \(\mu_1\) is the mean number of days on the disabled list in 2001, and \(\mu_2\) is the mean number of days in 1992.
• This assumption is tested against the alternative hypothesis to determine its validity based on the data.

Rejecting the null hypothesis suggests that an increase in mean disabled list time occurred, which is crucial for making data-driven decisions.

alternative hypothesis

Contrary to the null hypothesis, the alternative hypothesis (\(H_a\)) proposes that a change or difference does exist. In our baseball scenario, this would mean that players in 2001 spent more time on the disabled list compared to those in 1992.

• The alternative hypothesis is formulated as \(H_a: \mu_1 > \mu_2\), indicating the mean in 2001 is greater.
• This hypothesis is accepted if the statistical test provides enough evidence to reject the null hypothesis.
• It drives the research focus, revealing insights and conclusions about the observed data.

The acceptance of the alternative hypothesis could lead league officials to investigate reasons for increased injury recovery times.

percentage increase

Percentage increase provides a way to express the relative difference in values over time as a percentage. Observing injured Major League Baseball players spending more days on the disabled list in 2001 compared to 1992 involves calculating this percentage increase.

• We can find the percentage increase using the formula: \[ \text{Percentage Increase} = \left(\frac{\bar{x}_1 - \bar{x}_2}{\bar{x}_2}\right) \times 100 \]
• For our dataset, \(\bar{x}_1 = 60\) and \(\bar{x}_2 = 51\), leading to an increase of approximately 17.65%.
• This percentage shows that players on average spent significantly more time on the disabled list in 2001.

Calculating percentage increase helps communicate the scale of change in a straightforward manner, aiding in financial, planning, or operational decisions.