To determine if the proportion of women over 40 who regularly have mammograms is significantly different from 65%, we perform a hypothesis test using the z-test for proportions. The hypotheses for this two-tailed test are:
- Null hypothesis, [tex]\( H_0: p = 0.65 \)[/tex]
- Alternative hypothesis, [tex]\( H_a: p \ne 0.65 \)[/tex]
We are given the test statistic:
[tex]\[ z = -2.613 \][/tex]
To find the p-value associated with this z-score, we need to consult the standard normal distribution. The p-value for a two-tailed test is the probability of observing a test statistic as extreme or more extreme than the one observed under the null hypothesis.
First, we find the cumulative distribution function (CDF) value for [tex]\( z = -2.613 \)[/tex]. This value gives us the probability that a standard normal random variable is less than or equal to -2.613.
Next, because this is a two-tailed test, we need to account for both tails of the distribution. Hence, we multiply this CDF value by 2 to obtain the total p-value.
The cumulative probability for [tex]\( z = -2.613 \)[/tex] gives a p-value:
[tex]\[ p \approx 0.004487565111587978 \][/tex]
Multiplying this by 2 to account for both tails:
[tex]\[ p \approx 2 \times 0.004487565111587978 \][/tex]
[tex]\[ p \approx 0.008975130223175956 \][/tex]
Rounding to four decimal places, the p-value is approximately:
[tex]\[ p \approx 0.009 \][/tex]
Therefore, the p-value is:
[tex]\[ \boxed{0.009} \][/tex]
