Question

Q1: If X~(42,10) and  is computed from a random sample of size n=81, what is the distribution of  ?

Q2: If X~N(42,10) and  is computed from a random sample of size n=16, what is the distribution of  ?

Q3: When constructing a confidence interval for a mean, what are the two fundamentally different scenarios we would be working under?

Q4: Interpret the following probability statement into a complete sentence: P(x-bar > 20.26) = 0.8084

Q5: Find the following probability: P( Z > 0).

Answer 1

Solution:

1) X~(42,10) and  is computed from a random sample of size n=81.

Here n > 30, the Central Limit Theorem tells us that the sampling distribution will approximate a normal distribution.

So the distribution is Normal distribution .

2) X~N(42,10) and  is computed from a random sample of size n=16.

Here, population is normal, according to the Central limit theorem distribution of sample mean is normal.

So the distribution is Normal distribution.

3) When constructing a confidence interval for a mean, the two fundamentally different scenarios we would be working under are :

1) When population standard deviation is unknown and sample size is less than 30, then we use t distribution.

2) When population standard deviation is known and sample size is greater than 30, we use z distribution.

4) P(x-bar > 20.26) = 0.8084

The probability of the mean 20.26 is 0.8084

5) P(Z > 0) = 1 - P(Z < 0) = 1- 0.5000 = 0.5000 .... (From z table)

So P(Z > 0) = 0.5000