Scores for men on the verbal portion of the SAT-I test are normally distributed with a mean of 509 and a standard deviation of 112.
(a) If 1 man is randomly selected, find the probability that his score is at least 579.
(b) If 13 men are randomly selected, find the probability that their mean score is at least 579.
To solve this problem, we need to use the properties of the normal distribution. Here's how you can find the probabilities:
(a) To find the probability that a randomly selected man's score is at least 579, we need to find the area under the curve to the right of 579. We can use the formula for the standard normal distribution (Z-distribution) to convert the raw score to a Z-score and then find the corresponding probability.
First, we calculate the Z-score:
Z = (X - μ) / σ,
where X is the raw score, μ is the mean, and σ is the standard deviation.
In this case, X = 579, μ = 509, and σ = 112.
Z = (579 - 509) / 112
= 0.625
Next, we use a Z-table or a calculator to find the probability associated with this Z-score. The Z-table provides the area under the curve to the left of a given Z-score.
Looking up the Z-score of 0.625 in the Z-table, we find that the probability is approximately 0.7340.
So, the probability that a randomly selected man's score is at least 579 is 0.7340 or 73.40%.
(b) To find the probability that the mean score of 13 randomly selected men is at least 579, we need to use the Central Limit Theorem. According to the Central Limit Theorem, even if the underlying distribution is not normal, the distribution of the sample means will be approximately normal if the sample size is large enough.
The mean score of the sample means, also known as the sampling distribution of the mean, is equal to the population mean, which is 509 in this case.
The standard deviation of the sampling distribution of the mean, also known as the standard error, can be calculated using the formula:
σM = σ / √n,
where σ is the population standard deviation and n is the sample size.
In this case, σ = 112 and n = 13.
σM = 112 / √13
≈ 30.79
Now, we calculate the Z-score for the mean score of 579 using the formula:
Z = (X - μ) / σM,
where X is the mean score and μ is the population mean.
Z = (579 - 509) / 30.79
≈ 2.27
Using a Z-table or calculator, we can find the probability associated with this Z-score. The Z-table provides the area under the curve to the left of a given Z-score.
Looking up the Z-score of 2.27 in the Z-table, we find that the probability is approximately 0.9886.
So, the probability that the mean score of 13 randomly selected men is at least 579 is approximately 0.9886 or 98.86%.