Question

You would like to determine if more than 50% of the observations in a population are below 10. At α = 0.05, conduct the test on the basis of the following 20 sample observations: 8 12 5 9 14 11 12 6 8 9 2 6 11 9 3 7 8 4 13 10 Calculate the sample proportion.

Answer 1

There is not enough evidence to support the claim that more than 50% of the observations in a population are below 10

Explanation:

We are given the following in the question:

8, 12, 5, 9, 14, 11, 12, 6, 8, 9, 2, 6, 11, 9, 3, 7, 8, 4, 13, 10

Sample size, n = 20

p = 50% = 0.50

Alpha, α = 0.05

Observations below 10,x = 13

First, we design the null and the alternate hypothesis

`H_(0): p \leq 0.50\\H_A: p > 0.50`

This is a one-tailed(right) test.

Formula:

`\hat{p} = (x)/(n) = (13)/(20) = 0.65`

`z = \frac{\hat{p}-p}{\sqrt{(p(1-p))/(n)}}`

Putting the values, we get,

`z = \displaystyle\frac{0.65-0.50}{\sqrt{(0.50(1-0.50))/(20)}} = 1.3416`

Now, we calculate the p-value from the table.

P-value = 0.08986

Since the p-value is greater than the significance level, we fail to reject the null hypothesis and accept the null hypothesis.

Thus, there is not enough evidence to support the claim that more than 50% of the observations in a population are below 10