Final answer:
This answer addresses statistical concepts related to the normal distribution, such as hypothesis testing, z-scores, and sample size estimation, in the context of analyzing heights in different populations.
Step-by-step explanation:
The questions presented revolve around the concept of the normal distribution and its application in understanding and analyzing the height of individuals, particularly high school basketball players or other groups. Various aspects of statistics such as mean, standard deviation, z-scores, hypothesis testing, and the calculation of sample sizes for confidence intervals are covered.
Hypothesis Testing and p-value Interpretation
To determine if the mean height is less than 73 inches, the null hypothesis (H0) states that the mean height is 73 inches (${\mu} = 73$), while the alternative hypothesis (H1) suggests it is less than 73 inches (${\mu} < 73$). Given that the p-value is almost zero, we reject the null hypothesis, indicating strong evidence that the mean height is indeed less than 73 inches.
Calculating Z-scores
Z-scores indicate the number of standard deviations a data point is from the mean. For instance, a height of 77 inches has a z-score of -0.5141 indicating it is 0.5141 standard deviations below the mean; this height is considered shorter than average for NBA players. Conversely, a player with an 85-inch height has a z-score of 1.5424, meaning it is 1.5424 standard deviations above the mean and is considered taller than average.
Estimating Sample Sizes
To estimate a mean height within a desired precision or to construct a confidence interval, it is essential to calculate the appropriate sample size. Statistical formulas are used to determine the number of individuals that should be included in the sample based on the desired confidence level and margin of error.