To determine whether the proportion of women over 40 who regularly have mammograms is significantly less than 0.12, we are conducting a left-tailed test. The z-score for our sample data is [tex]\( z = -3.254 \)[/tex].
To find the p-value for a given z-score in a standard normal distribution (which is what we're using for this kind of hypothesis test), we utilize the cumulative distribution function (CDF) of a standard normal distribution. The CDF gives us the probability that a standard normal random variable is less than or equal to a given value.
For a left-tailed test:
1. Identify the test statistic (z-score): The given z-score is [tex]\( z = -3.254 \)[/tex].
2. Find the cumulative probability associated with the z-score: This is done using the standard normal CDF. The CDF will give us the probability that a standard normal variable is less than [tex]\( z \)[/tex].
By referring to the CDF tables or using statistical software, we can find the cumulative probability for [tex]\( z = -3.254 \)[/tex]. This value represents the p-value for our left-tailed test.
After looking up or calculating this cumulative probability, we find:
[tex]\[ \text{p-value} = 0.0006 \][/tex]
This p-value indicates the probability of observing a test statistic as extreme as, or more extreme than, the one observed under the null hypothesis of proportion of women over 40 who regularly have mammograms being 0.12 or greater.
Thus, the p-value accurate to 4 decimal places is:
[tex]\[ \text{p-value} = 0.0006 \][/tex]
