An April 15, 2002 report in Time Magazine stated that the average age for women to marry in the United States is now 25.9 years of age. If the standard deviation is assumed to be 4.3 years, find the probability that a random sample of 32 U.S. women would show a mean age at marriage of less than or equal to 24.9 years. (Give your answer correct to four decimal places.)
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the z score.
To find the probability that a random sample of 32 U.S. women would show a mean age at marriage of less than or equal to 24.9 years, we will use the Central Limit Theorem and the standard error.
The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
We can use the formula for the standard error of the mean (SE) to calculate the standard deviation of the sample mean, which is given by:
SE = standard deviation / √sample size
In this case, the standard deviation is 4.3 years and the sample size is 32. So, the standard error is:
SE = 4.3 / √32 = 0.76
Next, we need to standardize the sample mean using the z-score formula, which is:
z = (sample mean - population mean) / SE
In this case, the population mean is 25.9 years and the sample mean we want to find the probability for is 24.9 years. Using the formula, the z-score is:
z = (24.9 - 25.9) / 0.76 ≈ -1.32
Now, we can use a standard normal distribution table or calculator to find the probability associated with a z-score of -1.32. This probability represents the area under the curve to the left of -1.32.
Looking up the z-score in a standard normal distribution table, we find that the probability is approximately 0.0934.
Therefore, the probability that a random sample of 32 U.S. women would show a mean age at marriage of less than or equal to 24.9 years is approximately 0.0934 (or 0.0934 to four decimal places).