Can someone look at this picture and me? explain your answer aswell

1. b. 80; 5

2. c. 6

3. A. p = 0.0062

Step-by-step explanation:

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

Part 1

Let X the random variable who represents the variable of interest. We know from the problem that the distribution for the random variable X is given by:

We select samples of size n=16. That represent the sample size.  

From the central limit theorem we know that the distribution for the sample mean is given by:

So the expected value would be 80 and the standard error Se=20/sqrt(16)=5.

b. 80; 5

Part 2

Let X the random variable who represents the variable of interest. We know from the problem that the distribution for the random variable X is given by:

We select samples of size n=9. That represent the sample size.  

From the central limit theorem we know that the distribution for the sample mean is given by:

So the expected value would be 80 and the standard error Se=18/sqrt(9)=6.

c. 6

Part 3

From the central limit theorem we know that the distribution for the sample mean is given by:

And we want to calculate the probability that the mean for this sample would be greater than 75. So we want to find:

We can use the z score formula given by: