Question

Speed, size, and strength are thought to be important factors in football performance. The article "Physical and Performance Characteristics of NCAA Division I Football Players" (Research Quarterly for Exercise and Sport \([1990]: 395-401\) ) reported on physical characteristics of Division I starting football players in the 1988 football season. Information for teams ranked in the top 20 was easily obtained, and it was reported that the mean weight of starters on top- 20 teams was \(105 \mathrm{~kg}\). A random sample of 33 starting players (various positions were represented) from Division I teams that were not ranked in the top 20 resulted in a sample mean weight of \(103.3 \mathrm{~kg}\) and a sample standard deviation of \(16.3 \mathrm{~kg}\). Is there sufficient evidence to conclude that the mean weight for nontop- 20 starters is less than 105, the known value for top20 teams?

Short answer

Without the exact P-value, an exact answer cannot be given. However, it's determined by comparing the P-value against a 0.05 significance level. If P-value < 0.05, then the null hypothesis is rejected, suggesting that non-top 20 starters weigh less. If P-value > 0.05, then there's insufficient evidence to conclude that non-top 20 starters weigh less than top-20 starters.

Step-by-step solution

Identify the Hypotheses

We have the null hypothesis \(H_0: \mu = 105 kg\) suggesting that the average weight of non-top 20 starters equals that of top-20 starters. Conversely, the alternative hypothesis \(H_a: \mu < 105 kg\) assumes that non-top 20 starters weigh less than top-20 starters.

Calculate the Test Statistic

The test statistic (t) can be calculated using the formula: \[ t = \frac{{\bar{x} - \mu}}{{s / \sqrt{n}}}\] with \(\bar{x}\) being the sample mean, \(\mu\) the population mean, s the standard deviation of the sample, and n the sample size. In this case, \(\bar{x} = 103.3 kg, \mu = 105 kg, s = 16.3 kg, n = 33\). Substituting the values, we find the test statistic to be \(t = \frac{{103.3 - 105}}{{16.3 / \sqrt{33}}}\).

Find the P-Value

With the test statistic obtained, proceed to find the P-value using the t-distribution table. Due to the nature of the hypothesis test (one-tailed), locate the area to the left of the t value.

Determine Significance and Make a Conclusion

Once the P-value is calculated, compare it with a significance level (usually 0.05). If the P-value is less than the significance level, reject the null hypothesis and conclude that non-top 20 starters weigh significantly less than top 20 starters. If the P-value is higher, do not reject the null hypothesis implying that the weight difference is statistically insignificant.

Key concepts

Null Hypothesis

In statistical hypothesis testing, the null hypothesis, often symbolized as \(H_0\), is a statement positing that there is no effect or no difference, serving as the initial assumption to be tested. It represents the default or status quo condition that the test aims to challenge. For instance, in the context of football players' weights, the null hypothesis might assert that the mean weight of starting players on lower-ranked teams is equal to the standard weight of top players, symbolically \(H_0: \ = 105\) kg. This premise establishes a basis for comparison, where any noticeable deviation from this hypothesized average will be subject to rigorous examination through statistical methods.

Understanding the role of null hypothesis is crucial because it creates a benchmark for evaluating the presence of statistically significant evidence. In this way, the null hypothesis operates much like a skeptic's assertion; it will remain assumed true unless the data collected provides strong enough evidence to disprove it.

Alternative Hypothesis

The alternative hypothesis, denoted as \(H_a\) or \(H_1\), encapsulates the outcome that the study is set to prove or expect based on the research question. It is considered the counterpart to the null hypothesis and reflects a researcher's belief about the population parameter that should be detected if the null is not true. In the context of the football players' exercise, the alternative hypothesis contends that the average weight of non-top 20 starting players is less than that of top-tier teams, proposed as \(H_a: \ < 105\) kg. This hypothesis is not just a shot in the dark; it's informed by preliminary observations, theory, or prior evidence that suggest a potential difference.

The significance of articulating a clear alternative hypothesis lies in its ability to guide the direction of statistical testing. It determines the nature of the test to be one-tailed or two-tailed and sets the stage for either rejecting the null hypothesis or failing to find sufficient evidence against it. Students should grasp that the alternative hypothesis represents a new claim aiming to supplant the old one upon validation by data.

Test Statistic

Central to hypothesis testing is the test statistic, a standardized value derived from sample data which allows comparing the observed results with the distribution expected under the null hypothesis. This computed number conveys where the sample lies in relation to the null hypothesis. The test statistic used in the exercise is the t-statistic, following formula: \[ t = \frac{{\bar{x} - \}}{{s / \sqrt{n}}} \] where \(\bar{x}\) is the sample mean, \(\mu\) the hypothesized population mean, s the sample standard deviation, and n the number of observations in the sample.

This mathematical tool is pivotal as it accounts for sample variability and size, enabling researchers to infer about the population from their sample. The calculation of the test statistic in the exercise example is the very first numerical step in the process that leads to a decision about the hypotheses. It is a quantification of the observed effect (or difference) that serves as the basis for subsequent evaluation against possible outcomes under the null hypothesis.

P-Value

The p-value is a probability score that reflects the strength of the evidence against the null hypothesis. Its essence lies in indicating how extreme the test statistic is by providing the probability of observing a result as extreme as, or more extreme than, the one in the study, assuming that the null hypothesis is true. If this p-value is less than a predetermined significance level (commonly set at 0.05), it points to the data being unusual enough to cast doubt on the null hypothesis. In the exercise, once the t-statistic is determined, the next step involves finding out how likely it is to occur by referring to the t-distribution chart.

Interpreting the p-value is critical for students as it helps them make data-driven decisions. A low p-value signals that the observed data is unlikely under the assumption of the null, thus providing grounds to reject it in favor of the alternative. Conversely, a high p-value suggests that the data is consistent with the null hypothesis, leading to no rejection. Therefore, the p-value serves as a bridge between the calculated statistic and the hypothesis conclusion.

T-Distribution

When samples are small and population standard deviation is unknown, the t-distribution comes into play as a key component in hypothesis testing. It is a probability distribution resembling a normal distribution but with fatter tails, allowing for the added variability that arises from using the sample standard deviation as an estimate for the population standard deviation. The precise shape of the t-distribution depends on the degrees of freedom, which is related to the sample size (usually number of observations minus one).

In the context of the weight comparison among football players, the t-distribution helps determine the likelihood of observing a sample mean as extreme as 103.3 kg, given the null hypothesis that the true mean is 105 kg. Students should understand the use of the t-distribution in the context of small samples where the z-distribution (used for larger samples with a known population standard deviation) is not suitable. It represents a practical tool for interpreting the test statistic in relation to what we might expect if the null hypothesis holds true.