The length of pregnancies are normally distributed with a mean of 273 days in the standard deviation the 15 days. If 36 women are randomly selected find the probability that they have a mean pregnancy between 273 days in 275 day
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To find the probability that the sample mean pregnancy length falls between 273 and 275 days, we can use the Central Limit Theorem. According to the Central Limit Theorem, the distribution of sample means will be approximately normal, regardless of the shape of the population distribution, as long as the sample size is large enough (usually a minimum of 30 observations).
In this case, since you have a sample of 36 women, we can assume that the sample mean pregnancy length will be approximately normally distributed.
First, we need to standardize the sample mean using the formula for the standard error of the mean:
Standard Error of the Mean (SE) = standard deviation / √(sample size)
SE = 15 / √(36)
= 15 / 6
= 2.5
Next, we need to calculate the z-scores for the lower and upper bounds of the range:
Lower z-score = (273 - mean) / SE
= (273 - 273) / 2.5
= 0
Upper z-score = (275 - mean) / SE
= (275 - 273) / 2.5
= 0.8
Now, we can find the probability using a standard normal distribution table or a calculator.
The probability that the sample mean falls between 273 and 275 days can be found by finding the area between the lower and upper z-scores on the standard normal distribution curve.
P(273 ≤ X ≤ 275) = P(0 ≤ Z ≤ 0.8)
Using a standard normal distribution table or calculator, we find the probability corresponding to a z-score of 0.8 is 0.7881.
Therefore, the probability that the mean pregnancy length of the 36 women falls between 273 and 275 days is 0.7881 or 78.81%.