Respuesta :
Answer:
a) a. Normal
b) The mean of the sampling distribution of the sample mean is 40.
c) C. The standard error of the mean decreases.
d) The standard deviation of the sampling distribution of the sample mean is 2.58.
Step-by-step explanation:
(a) If a random variable X is normally distributed, what will be the shape of the distribution of the sample mean?
If the variable is normally distributed, then the sample mean is going to be normally distributed. So the correct answer is
a. Normal
(b) If the mean of a random variable X is 40, what will be the mean of the sampling distribution of the sample mean?
The mean of the sampling distribuion is the same, so it is 40.
(c) As the sample size n increases, what happens to the standard error of the mean?
We have that:
[tex]S_{E} = \frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation. So, as the sample size n increases, the standard error of the mean decreases. The correct answer is
C. The standard error of the mean decreases.
(d) If the standard deviation of a random variable X is 10 and a random sample of size nequals15 is obtained, what is the standard deviation of the sampling distribution of the sample mean
We have that:
[tex]s = \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{15}} = 2.58[/tex]
The standard deviation of the sampling distribution of the sample mean is 2.58.
A) The correct shape of the distribution of the sample mean is; Option B: Normal
B) If the mean of a random variable X is 40, the mean of the sampling distribution of the sample mean is; 40
C) As the sample size n increases, the standard error of the mean will;
Option C; The standard error of the mean decreases.
D) The standard deviation of the sampling distribution of the sample mean = 2.58
A) We are told that the Random variable is normally distributed. This means that the shape of the distribution curve will also be normal because the sample mean will be normal.
B) We are told that the mean of the Random variable is 40. This simply means that the sample mean is 40.
C) Formula of standard error of the mean is;
SE = σ/√n
Where;
σ is standard deviation
n is sample size
This means that the higher the sample size, the lower the Standard error and the also the lesser the sample the sample size, the greater the standard error.
Thus, option C is correct.
D) We are given that standard deviation of a random variable X is 10 and a random sample of size n equals 15.
Thus;
σ = 10
n = 15
Standard deviation of sample distribution = 10/√15 = 2.58
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