Question

In a gallup poll, 1010 of those who are randomly selected and asked if they were satisfied or dissatisfied with the amount of leisure time that they had period of this symbol 657 said that they were satisfied and 353 said that they were dissatisfied. Use a 0.01 significance level to test the claim that 2/3 of those are satisfied with the amount of leisure time that they have.

Answer 1

The 0.01 significance level, there is not enough evidence to conclude that the proportion of people satisfied with their leisure time is different from
`(2)/(3)` .

To test the claim that
`(2)/(3)` of those surveyed are satisfied with the amount of leisure time they have, we can set up a hypothesis test. Let p be the proportion of people who are satisfied.

The null hypothesis H_0 is that the proportion is equal to
`(2)/(3)` (no difference):

H_0 :p=0.67

The alternative hypothesis H_1 is that the proportion is not equal to
`(2)/(3)` (there is a difference):

H_1 :p=0.67

The significance level is

α=0.01.

We can use the z-test for proportions to test this hypothesis. The test statistic z is given by:

`z= \frac{p-p_o}{\sqrt{(p_0.(1-p_0))/(n) } }`

​where:

p is the sample proportion,p_0 is the hypothesized population proportion,

n is the sample size.

The critical values for a two-tailed test at a α=0.01 significance level are approximately ±2.58 (you can find these in standard normal distribution tables).

Let's calculate p, the standard error, and the test statistic:

`p= (657)/(1010) = 0.6505`

`standard error= \sqrt{(0.67(1-0.67))/(1010) } = 0.0153`

`z= (0.6505-0.67)/(0.0153) = -1.31`

Now, compare the absolute value of the test statistic with the critical value 2.58. Since ∣−1.31∣<2.58, we fail to reject the null hypothesis.

Therefore, at the 0.01 significance level, there is not enough evidence to conclude that the proportion of people satisfied with their leisure time is different from
`(2)/(3)` .