8. A sample of 65 observations is selected from one population. The sample mean is 2.67 and the sample standard deviation is .75. A sample of 50 observations is selected from a second population. The sample mean is 2.59 and the sample standard deviation is .66. Conduct the following test using 95% level of confidence. Z-test for Unpaired data
a. Is it a 1 or 2 tail test
b. Compute the statistical calculation
c. What is your decision regarding H1
d. What is the P-value
a. To determine whether the test is a one-tail or two-tail test, we need to look at the alternative hypothesis (H1). If H1 states that the means of the two populations are either greater than or less than each other (but not both), then it is a one-tail test. If H1 states that the means of the two populations are different from each other (greater than or less than), then it is a two-tail test.
b. To compute the statistical calculation, we need to calculate the test statistic, which, in this case, is the Z-test for unpaired data. The formula for the Z-test is:
Z = (sample mean 1 - sample mean 2) / √[(sample standard deviation 1² / sample size 1) + (sample standard deviation 2² / sample size 2)]
In this case, the sample mean 1 = 2.67, sample mean 2 = 2.59, sample standard deviation 1 = 0.75, sample standard deviation 2 = 0.66, sample size 1 = 65, and sample size 2 = 50. Plugging in these values into the formula, we can calculate the Z-test statistic.
c. To make a decision regarding H1, we compare the Z-test statistic to the critical value(s) based on the chosen level of confidence (95% in this case). If the Z-test statistic falls within the critical region, we reject the null hypothesis (H0) and accept H1. If the Z-test statistic falls outside the critical region, we fail to reject H0.
d. The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic, assuming H0 is true. To calculate the p-value, we need to use a standard normal distribution table or a statistical software. We compare the absolute value of the Z-test statistic to the critical values in the standard normal distribution table or use the software to find the corresponding p-value.
By comparing the calculated p-value to the significance level (alpha), usually set at 0.05 for a 95% confidence level, we can make a decision whether to reject or fail to reject H0. If the p-value is less than alpha, we reject H0. If the p-value is greater than alpha, we fail to reject H0.