Final answer:
The data support the claim that the mean height of high school basketball players is less than 73 inches due to a very low p-value. The most famous quarterback was approximately 0.79 standard deviations below the mean weight of his team, while a comparison Texas player was 0.70 standard deviations below their team's mean, making the quarterback lighter in terms of standard deviations. Z-scores are calculated for heights, with the interpretation that values 3 or more standard deviations from the mean are unusual.
Step-by-step explanation:
In order to determine if the data support the claim that the mean height of high school basketball players is less than 73 inches, we begin by stating our null and alternative hypotheses:
- H0 (null hypothesis): μ = 73 inches - This implies that the true population mean height is 73 inches.
- HA (alternative hypothesis): μ < 73 inches - This suggests that the true population mean height is less than 73 inches.
A p-value almost zero indicates that the probability of observing a sample mean of 71 inches or less is extremely low if the true population mean is 73 inches. Therefore, we reject the null hypothesis in favor of the alternative, supporting the claim that the mean height is indeed less than 73 inches.
As for the standard deviation of the team's most famous quarterback who weighed 205 pounds, we compare this weight to the mean weight of a Texas football player. With a mean of 240.08 pounds and a standard deviation of 44.38 pounds, the calculation would go as follows:
Z = (Player's Weight - Mean Weight) / Standard Deviation
Z = (205 - 240.08) / 44.38
Z ≈ -0.79
The quarterback's weight is approximately 0.79 standard deviations below the mean of his team. Comparatively, a Texas player who weighed 209 pounds would have a Z of (209 - 240.08) / 44.38 ≈ -0.70, which means this Texas player is about 0.70 standard deviations below the mean of their team. Given that -0.79 < -0.70, the California quarterback was lighter in terms of standard deviations from their respective team's mean weight.
The z-scores for the listed heights are as follows:
- 77 inches: Z = (77 - 79) / 3.89 ≈ -0.5141
- 85 inches: Z = (85 - 79) / 3.89 ≈ 1.5432
- Z-score of 3.5 is highly unlikely, indicating that a player reporting a height that far from the mean could be exaggerating or reporting inaccurately, as it is more than 3 standard deviations away from the mean, which is very rare in a normal distribution.