A random sample of 10 observations is selected from the first normal population and 8 from the second normal population. For the hypothesis, H0: ¦Ì1 ¡Ü ¦Ì2, (.01 significance level), what is (are) the critical value(s)? (Population standard deviations are unknown).
You need some measure of variability to answer this question.
A random sample of 10 observations is selected from the first normal population and 8 from the second normal population. For the hypothesis, H0: ¦Ì1 ¡Ü ¦Ì2, (.01 significance level), what is (are) the critical value(s)? (Population standard deviations are unknown
To find the critical value(s) for the hypothesis test, you need to calculate the degrees of freedom and use them to look up the critical value in a t-distribution table.
For this hypothesis test, where the population standard deviations are unknown, we need to use the pooled t-test. The formula to calculate the degrees of freedom in this case is:
df = (s1^2/n1 + s2^2/n2)^2 / [(s1^2/n1)^2 / (n1-1) + (s2^2/n2)^2 / (n2-1)]
Where:
- s1 and s2 are the sample standard deviations of the two populations, respectively.
- n1 and n2 are the sample sizes of the two populations, respectively.
In this case, the sample sizes are n1=10 and n2=8, but we don't have the sample standard deviations. So, we need to estimate them using the sample variances.
Next, you will need to calculate the sample variances using the formulas:
s1^2 = ((sum of (X - X1)^2) / (n1-1))
s2^2 = ((sum of (X - X2)^2) / (n2-1))
Where X1 and X2 are the sample means, respectively.
Once you have calculated the sample variances (s1^2 and s2^2), you can insert them into the degrees of freedom formula to find the degrees of freedom (df).
Now, with the degrees of freedom, you can look up the critical value(s) in a t-distribution table based on the given significance level (0.01).
The critical value(s) will depend on whether the test is one-tailed or two-tailed. You can determine this based on the alternative hypothesis (H1). If H1 is of the form: ¦Ì1 > ¦Ì2, or ¦Ì1 < ¦Ì2, then it is a one-tailed test. If H1 is of the form: ¦Ì1 ≠ ¦Ì2, then it is a two-tailed test.
Look up the critical value(s) in the t-distribution table for the appropriate degrees of freedom and significance level. The critical value for a two-tailed test would be split among the upper and lower tails of the distribution.
Note: If you have access to statistical software like R, you can also use functions like "qt()" or "pt()" to find the critical value(s) directly.