A cancer research group surveys 100 women more than 40 years old to test the hypothesis that 50% of women in this age group have regularly scheduled mammograms.Forty of the women respond affirmatively.Using $\alpha$ = 0.05,the critical value of chi-square is _______.

Question

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To find the critical value of chi-square, we need to perform a chi-square goodness-of-fit test. The null hypothesis is that 50% of women over 40 have regularly scheduled mammograms. The alternative hypothesis is that less than 50% (since we are testing for a one-tailed hypothesis).

First, we calculate the expected number of women who would have regularly scheduled mammograms if the null hypothesis were true. That would be:
Expected count = 100 * 0.5 = 50

We can now calculate the chi-square statistic:
chi-square = sum((observed count - expected count)^2 / expected count)
chi-square = ((40-50)^2 / 50) + ((60-50)^2 / 50)
chi-square = 4

Using a chi-square distribution table with (df=1) and (α=0.05), we find the critical value to be 3.84. Since our calculated chi-square (4) is greater than the critical value (3.84), we reject the null hypothesis and conclude that less than 50% of women over 40 have regularly scheduled mammograms.

A Cancer Research Group Surveys 100 Women More Than 40 α\alphaα

A Cancer Research Group Surveys 100 Women More Than 40 $\alpha$