social worker believes that fewer than 25% of couples in a certain area have ever used any form of birth control. A random sample of 120 couples was contacted. 20 of them said that they have used. Test the belief of the social worker at 0.05 level
To test the belief of the social worker, we can use a hypothesis test for a population proportion.
Null Hypothesis (H0): The proportion of couples who have ever used birth control is 0.25.
Alternative Hypothesis (H1): The proportion of couples who have ever used birth control is less than 0.25.
Let p be the population proportion of couples who have ever used birth control. We can set up the following hypothesis test using a one-tailed z-test:
H0: p = 0.25
H1: p < 0.25
We can calculate the test statistic using the formula:
z = (p̂ - p0) / sqrt(p0*(1-p0)/n)
Where:
p̂ = 20/120 = 0.1667 (proportion of sample couples who have used birth control)
p0 = 0.25 (population proportion assumed by the social worker)
n = 120 (sample size)
Plugging in the values, we get:
z = (0.1667 - 0.25) / sqrt(0.25 * (1-0.25) / 120)
z = (-0.0833) / sqrt(0.1875 / 120)
z = -0.0833 / 0.0433
z = -1.922
Using a z-table, we can find the critical value for a one-tailed test at a 0.05 significance level, which is approximately -1.645. Since our calculated test statistic (-1.922) is less than the critical value (-1.645), we reject the null hypothesis.
Therefore, based on the sample data, we have enough evidence to suggest that the proportion of couples who have ever used birth control in the given area is less than 25%, contradicting the belief of the social worker.