The average adult height of a particular species of kangaroo is 64 inches. However, it is conjectured that a group of such kangaroos living in a particular remote location of Australia are actually shorter which might indicate that they belong to a different subspecies. A sample of 37 adult kangaroos from this group is observed, and the sample mean is found to be 62.4 inches. There is evidence to assume that the heights are normally distributed with a population standard deviation of σ= 4.1 inches. Conduct a test of Ho: 64 vs. H1: µ < 64=”” at=”” the=”” 5%=”” significance=”” level.=”” determine=”” the=”” p-value=”” of=”” the=”” test,=”” and=”” state=”” your=””>
The Correct Answer and Explanation is :
To test the hypothesis that the kangaroos in this remote Australian location have a mean height less than the known average of 64 inches, we perform a one-sample z-test. The null hypothesis (H₀) states that the population mean height is 64 inches (H₀: μ = 64), while the alternative hypothesis (H₁) posits that the population mean height is less than 64 inches (H₁: μ < 64).
Given:
- Population mean under H₀ (μ₀): 64 inches
- Sample mean (x̄): 62.4 inches
- Population standard deviation (σ): 4.1 inches
- Sample size (n): 37
First, we calculate the standard error of the mean (SEM), which measures the dispersion of sample means around the population mean:
[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{4.1}{\sqrt{37}} \approx 0.674 ]
Next, we compute the z-score, which indicates how many standard errors the sample mean is away from the population mean:
[ z = \frac{x̄ – μ₀}{\text{SEM}} = \frac{62.4 – 64}{0.674} \approx -2.37 ]
A z-score of -2.37 signifies that the sample mean is approximately 2.37 standard errors below the hypothesized population mean.
To determine the p-value, we consult the standard normal distribution. The p-value represents the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. For a z-score of -2.37, the cumulative probability is approximately 0.0089. Since this is a one-tailed test (we are interested in whether the mean is less than 64 inches), the p-value is 0.0089.
At a significance level (α) of 0.05, we compare the p-value to α:
- If p-value ≤ α, we reject H₀.
- If p-value > α, we fail to reject H₀.
In this case, the p-value (0.0089) is less than the significance level (0.05), leading us to reject the null hypothesis. This suggests that there is statistically significant evidence at the 5% significance level to conclude that the mean height of kangaroos in this remote location is less than 64 inches.
It’s important to note that while this statistical test indicates a significant difference in mean height, it does not confirm the existence of a different subspecies. Further biological and genetic studies would be necessary to explore that possibility.