Answer to Question #306566 in Statistics and Probability for love

Question

A population consists of the numbers 4,7,10,11, and 13

1.enumerate all possible samples of size 2 without replacement

2.construct a sampling distribution table and compute the mean of each sample.

3. find the mean of the population .show your solution

4.find the variance of the population .show your solution.

5.Find the mean of the sampling distribution of the sample means .show your solution.

6.find the variance of the sampling distribution of the sample means.show your solution

Answer 1

1

The possible samples of size 2 without replacement are:

4,7,10,11 and 13

Sample space

{4,7}

(4,10}

{4,11}

{4,13}

{7,10}

{7,11}

{7,13}

{10,11}

{10,13}

{11,13}

2.

Let the random variable X represent the possible values of the sample means

Sample space Sample means (X)

{4,7} (4+7)/2 = 5.5

(4,10} (4+10)/2 = 7

{4,11} (4+11)/2 = 7.5

{4,13} (4+13)/2 = 8.5

{7,10} (7+10)/2=8.5

{7,11} (7+11)/2 = 9

{7,13} (7+13)/2 = 10

{10,11} (10+11)/2 = 10.5

{10,13} (10+13)/2 = 11.5

{11,13} (11+13) = 12

3.

Given that the population consists of the numbers 4,7,10,11, and 13

the population mean is the average of the 5 numbers (N=5)

Therefore, the mean of the population $\mu$=$\frac{1}{5}$( 4+7+10+11+13) = 9

Answer: Population mean $\mu$ =9

4.

The population variance is obtained by squaring the difference between each number and the average, we sum these together and divide the result by population size (N)

Thus'

$\sigma^{2}$ = $\frac{Σ(X-\mu)^{2}}{N}$

= $\frac{(4-9)^{2}+(7-9)^{2}+(10-9)^{2}+(11-9)^{2}+(13-9)^{2}}{5}$

= 10

Answer: Population variance $\sigma^{2}$= 10

5.

Sample space Sample means (X)

{4,7} (4+7)/2 = 5.5

(4,10} (4+10)/2 = 7

{4,11} (4+11)/2 = 7.5

{4,13} (4+13)/2 = 8.5

{7,10} (7+10)/2=8.5

{7,11} (7+11)/2 = 9

{7,13} (7+13)/2 = 10

{10,11} (10+11)/2 = 10.5

{10,13} (10+13)/2 = 11.5

{11,13} (11+13) = 12

The possible values of the random variable X are:

X = (5.5, 7, 7.5, 8.5, 9, 10, 10.5, 11.5, 12)

with probabilities

P(X) = (1/10, 1/10, 1/10, 2/10, 1/10,1/10,1/10,1/10, 1/10) respectively

The mean of the sampling distribution of the sample means is:

$\mu$_x = ΣX*P(X)

=5.5(1/10)+7(1/10)+7.5(1/10)+8.5(2/10)+9(1/10) + 10(1/10)+10.5(1/10) + 11.5(1/10) +12(1/10)

=9

Answer: $\mu$_x = 9

6.

The variance of the sampling distribution of the sample means.

σ_x² = Σ(x – μ_x)²⋅ P(x)

=$\frac{1}{10}$(5.5-9)² + $\frac{1}{10}$(7-9)² + $\frac{1}{10}$(7.5-9)²+ $\frac{2}{10}$(8.5-9)² + $\frac{1}{10}$(9-9)² + $\frac{1}{10}$(10-9)² +$\frac{1}{10}$(10.5-9)²+ $\frac{1}{10}$(11.5-9)²+ $\frac{1}{10}$(12-9)² 

Answer: σ_x² = 3.75