Actual lengths of pregnancy terms are nearly normally distributed about a mean pregnancy length (of about 38-39 weeks) with a standard deviation of 15 days: about what percentage of births would be expected to occur within 15 days of the mean pregnancy length?
To determine the percentage of births expected to occur within 15 days of the mean pregnancy length, you need to calculate the area under the normal distribution curve within that range.
First, let's find the z-scores for the upper and lower boundaries of the range. The z-score formula is given as:
z = (x - μ) / σ
Where:
x = the value you want to find the z-score for (in this case, 15 days)
μ = the mean of the distribution (38-39 weeks, which can be converted to days)
σ = the standard deviation (15 days)
To convert the weeks to days, we should multiply by 7 (since there are 7 days in a week):
Lower Bound: (38 weeks * 7 days/week) = 266 days
Upper Bound: (39 weeks * 7 days/week) = 273 days
Now, let's calculate the z-scores:
Lower Bound z-score: (266 - 273) / 15 = -7/15 = -0.467
Upper Bound z-score: (273 - 273) / 15 = 0
Now we have the z-scores for the lower and upper bounds.
Next, we need to consult the Z-table (a table that shows the area under the standard normal distribution curve for different z-scores). The Z-table provides the cumulative probability from the leftmost side up to the given z-score.
Using the Z-table, we can find the percentage of births expected to occur within 15 days of the mean pregnancy length. By looking up the z-scores we calculated, we can find the corresponding probabilities.
The z-table tells us that:
For Lower Bound z-score -0.467: the cumulative probability is 0.3192 (or approximately 31.92%)
For Upper Bound z-score 0: the cumulative probability is 0.5000 (or 50.00%)
Now, to find the percentage within the range, subtract the lower bound probability from the upper bound probability:
Percentage = Upper Bound Probability - Lower Bound Probability
= 50.00% - 31.92%
= 18.08%
Therefore, about 18.08% of births would be expected to occur within 15 days of the mean pregnancy length.