EXAMPLE 4 Length of human pregnancies

Researchers have recently begun to question the actual length of a human pregnancy. According to a report published on August 6, 2013, although women are typically given a delivery date that is calculated as 280 days after the onset of their last menstrual period, only 4% of women deliver babies at 280 days. A more likely average time from ovulation to birth may be much less than 280 days. To test this theory, a random sample of 95 women with healthy pregnancies is monitored from ovulation to birth. The mean length of pregnancy is found to be days, with a standard deviation of s = 10 days. Is this sample result good evidence that the mean length of a healthy pregnancy for all women is less than 280 days?

The hypotheses. The researcher’s claim is that the mean length of pregnancy is less than 280 days. That’s our alternative hypothesis, the statement we seek evidence for. The hypotheses are

H₀: μ = 280

H_a: μ < 280

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The sampling distribution. If the null hypothesis is true, the sample mean has approximately the Normal distribution with mean μ = 280 and standard error

We once again use the sample standard deviation s in place of the unknown population standard deviation σ.

The data. The researcher’s sample gave . The standard score, or test statistic, for this outcome is

That is, the sample result is about 4.85 standard errors below the mean we would expect if, on the average, healthy pregnancies lasted for 280 days.

The P-value. Figure 22.4 locates the sample outcome −4.85 (in the standard scale) on the Normal curve that represents the sampling distribution if H₀ is true. This curve has mean 0 and standard deviation 1 because we are using the standard scale. The P-value for our one-sided test is the area to the left of −4.85 under the Normal curve. Figure 22.4 indicates that this area is very small. The smallest value in Table B is −3.4 and Table B says that −3.4 is the 0.03 percentile, so the area to its left is 0.0003. Because −4.85 is smaller than −3.4, we know that the area to its left is smaller than 0.0003. Thus, our P-value is smaller than 0.0003.

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The conclusion. A P-value of less than 0.0003 is strong evidence that the mean length of a healthy human pregnancy is below the commonly reported length of 280 days.