Respuesta :
Answer:
We conclude that the population mean is greater than 10.
Step-by-step explanation:
The complete question is: A random sample of 10 observations is selected from a normal population. The sample mean was 12 and the sample standard deviation was 3. Using the 0.05 significance level: a. State the decision rule. b. Compute the value of the test statistic. c. What is your decision regarding the null hypothesis [tex]H_0= \mu \leq 10[/tex] and [tex]H_A=\mu >10[/tex].
We are given that a random sample of 10 observations is selected from a normal population. The sample mean was 12 and the sample standard deviation was 3.
Let [tex]\mu[/tex] = population mean
So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \leq 10[/tex] {means that the population mean is less than or equal to 10}
Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] > 10 {means that the population mean is greater than 10}
The test statistics that will be used here is One-sample t-test statistics because we don't know about the population standard deviation;
T.S. = [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex] ~ [tex]t_n_-_1[/tex]
where, [tex]\bar X[/tex] = sample mean = 12
s = sample standard deviation = 3
n = sample of observations = 10
So, the test statistics = [tex]\frac{12-10}{\frac{3}{\sqrt{10} } }[/tex] ~ [tex]t_9[/tex]
= 2.108
The value of t-test statistics is 2.108.
Now, at a 0.05 level of significance, the t table gives a critical value of 1.833 at 9 degrees of freedom for the right-tailed test.
Since the value of our test statistics is more than the critical value of t as 2.108 > 1.833, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.
Therefore, we conclude that the population mean is greater than 10.
