Insights into the concepts of hypothesis testing (Part 5)

Understanding Steps of Hypothesis Testing

In continuation to yesterday’s discussion, a sample of size 5 is drawn from a normally distributed population with population variance equal to 0.75. The sample mean is 5.112. Test the hypothesis at 5% level of significance that this sample comes from a population with mean 5.

1.       Null Hypothesis: Population Mean = 5

(Parent population of this sample has a mean of 5)

2.       Alternative Hypothesis: Population Mean >5

(Parent population of this sample has a mean of greater than 5. Since sample mean is 5.112, we think that alternative hypothesis might be true)

3.       Alpha = 0.05

(Probability of error of rejecting a true null hypothesis is 5 in 100)

4.       Z = (sample mean - population mean (under null hypothesis))/standard error of sample mean 

Sample mean = 5.112. Big Question- Is this sample mean large enough to conclude that it comes from parent population of Mean > 5

We know that sample mean of 5.112 is greater than the population mean of 5. But we are generalizing results from a sample of size 5 to the entire population that is we are inferring for the whole population so these steps of hypothesis testing have to be followed. The figure below gives the sampling distribution of sample mean. We see that the sample mean should be at least 5.634 for us to conclude that the sample mean is large enough for us to conclude the null hypothesis.
Sampling Distribution of Sample Mean with Mean Equal to 5 and Standard Error Equal to 0.385