Final answer:
If the t statistic was 4.56, it would typically lead to rejection of the null hypothesis due to significance. For a chi-square distribution with 20 degrees of freedom, the mean is 20 and the standard deviation is approximately 6.32. Chi-square distribution becomes more symmetrical with more degrees of freedom, but its standard deviation isn't twice the mean.
Step-by-step explanation:
Understanding the Chi-Square Distribution and t-Statistic Interpretation
If the t statistic had been 4.56, it would imply that the observed sample mean is significantly different from the hypothesized population mean, assuming normal distribution. Given a high value like 4.56, this would typically lead to the rejection of the null hypothesis at a conventional significance level (e.g., α = 0.05), as the t statistic would be in the critical region where the probability of observing such a value by chance is very low.
Regarding the chi-square distribution, for 20 degrees of freedom, the mean (μ) would be equal to the degrees of freedom, which is 20. The standard deviation (σ) is calculated as the square root of twice the degrees of freedom, which would be the square root of 40, approximately 6.32. Therefore, the chi-square distribution with 20 degrees of freedom has a mean of 20 and a standard deviation of approximately 6.32.
When assessing the characteristics of the chi-square distribution, the statements are as follows:
- The graph of the chi-square distribution becomes more symmetrical as the number of degrees of freedom increases, which is true.
- The standard deviation of the chi-square distribution is twice the mean, which is false. The standard deviation is the square root of twice the mean.
- If df = 24, the mean and median of the chi-square distribution are the same, which is false; the mean and median are different for all non-normal distributions.