To find the t-statistic for the given percentiles, we need to know the sample mean, sample standard deviation, and degrees of freedom (df). Assuming that we are dealing with a sample of size 55, we can use the t-distribution to calculate the t-statistics.
(a) To find the t-statistic for the 45th percentile, we first need to find the corresponding t-score. Using a t-distribution table or a calculator, we can find that the t-score for the 45th percentile with 54 degrees of freedom is approximately -0.1812.
Next, we can use the formula for calculating the t-statistic:
t = (x - μ) / (s / √n)
where x is the sample percentile, μ is the population mean (which is unknown), s is the sample standard deviation, and n is the sample size.
Since we don't know the population mean, we can use the sample mean as an estimate. Let's assume that the sample mean is 10 and the sample standard deviation is 2. Then, the t-statistic for the 45th percentile can be calculated as:
t = (x - μ) / (s / √n) = (0.45 - 10) / (2 / √55) ≈ -10.03
Therefore, the t-statistic for the 45th percentile is approximately -10.03.
(b) To find the t-statistic for the 95th percentile, we first need to find the corresponding t-score. Using a t-distribution table or a calculator, we can find that the t-score for the 95th percentile with 54 degrees of freedom is approximately 1.6759.
Next, we can use the same formula for calculating the t-statistic:
t = (x - μ) / (s / √n)
Assuming that the sample mean is still 10 and the sample standard deviation is 2, the t-statistic for the 95th percentile can be calculated as:
t = (x - μ) / (s / √n) = (0.95 - 10) / (2 / √55) ≈ -26.97
Therefore, the t-statistic for the 95th percentile is approximately -26.97.