You wish to test the following claim (Ha) at a significance level of α=0.001
Ho:μ1=μ2
Ha:μ1<μ2
You believe both populations are normally distributed, but you do not know the standard deviations for either. We will assume that the population variances are not equal.
You obtain a sample of size n1=17 with a mean of M1=70.5 and a standard deviation of SD1=10.7 from the first population. You obtain a sample of size n2=14 with a mean of M2=79.6 and a standard deviation of SD2=8.7 from the second population.
What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =
What is the p-value for this sample? For this calculation, use the conservative under-estimate for the degrees of freedom. The degrees of freedom is the minimum of n1 - 1 and n2 - 1. (Report answer accurate to four decimal places.)
p-value =
To test the claim Ho:μ1=μ2 against Ha:μ1<μ2, we need to perform a two-sample t-test.
The formula for the test statistic in this case is given by:
t = (M1 - M2) / sqrt((SD1^2 / n1) + (SD2^2 / n2))
Plugging in the given values:
t = (70.5 - 79.6) / sqrt((10.7^2 / 17) + (8.7^2 / 14))
Calculating this expression, we get:
t = -3.184
So, the test statistic for this sample is -3.184 (accurate to three decimal places).
To find the p-value, we first need to determine the degrees of freedom. The degrees of freedom for the t-test is the minimum of (n1 - 1) and (n2 - 1). In this case, the degrees of freedom is the minimum of (17 - 1) and (14 - 1), which is 13.
Next, we calculate the p-value using the t-distribution with the degrees of freedom we found. Since the alternative hypothesis is one-tailed (μ1 < μ2), we want to find the probability of observing a more extreme value than the test statistic in the direction of the alternative hypothesis.
From the t-distribution table or using statistical software, we find that the p-value for a t-statistic of -3.184 with 13 degrees of freedom is approximately 0.0042.
Therefore, the p-value for this sample is 0.0042 (accurate to four decimal places).