Central Limit Theorem: How do you calculate the test statistic?

Paper: 6691/01R Edexcel GCE Statistics S3 June 2014R

Hello! I'm wondering why the test statistic's denominator is 6/(square root of 5). The central limit theorem says population variance is equal to the sample variance, but why do we sometimes have the denominator of the square root of (population variance and sample variance)?

Question 4: A manufacturing company produces solar panels. The output of each solar panel is normally distributed with a standard deviation of 6 watts. It is thought that the mean output is 160 watts.
A researcher believes that the mean output of the solar panels is greater than 160 watts. He writes down the output values of 5 randomly selected solar panels. He uses the data to carry out a hypothesis test at the 5% level of significance. He tests H0: mu= 160 against H1: mu > 160
On reporting to his manager, the researcher can only find 4 of the output values. These are shown below:
168.2 157.4 173.3 161.1
Given that the result of the hypothesis test is that there is significant evidence to reject H0 at the 5% level of significance, calculate the minimum possible missing output value.

It's nothing to do with the central limit theorem. It's not about comparing sample and population distributions.

When you estimate the mean of the distribution, the reliability increases when you collect more data. The variance of the mean estimate is sigma^2/n, where sigma^2 is the variance of the original distribution. The random variable here is the mean estimate.

Hi,
For the same question, i dont understand how they got X in the test statistic (132+alpha/5)

but where did they get 132+alpha/5 from

ohhh so the sample mean is (660 + alpha)/5 which simplifies to 132+ alpha/5. Thats great! but where did they get 1.6449 from, i know it's z.

Im not sure if this is useful, but i still dont underneayh

I just dont get why its more than 1.6449, as i put less than 0.05 since that is the significance level to reject the original hypothesis.