Assume that the population of heights of male college students is approximately normally distributed with mean of 72.83 inches and standard deviation of 6.86 inches. A random sample of 93 heights is obtained. Show all work.
(A) Find the mean and standard error of the x distribution
(B) Find p(x > 73.25)
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To answer the given questions, we first need to understand some basic concepts.
(A) Finding the mean and standard error of the x distribution:
1. The mean, also known as the average, of a distribution is denoted by μ (mu). In this case, the mean height of male college students is given as 72.83 inches.
2. The standard deviation, denoted by σ (sigma), measures the spread of the distribution. For this population, the standard deviation is given as 6.86 inches.
3. The standard error (SE) of the x distribution is calculated as the standard deviation of the population divided by the square root of the sample size.
Now let's calculate the mean and standard error:
Mean (μ) = 72.83 inches
Standard Error (SE) = σ / √n, where σ is the standard deviation and n is the sample size.
Given that the sample size is 93 and the standard deviation is 6.86, we can calculate the standard error:
SE = 6.86 / √93 ≈ 0.712 (rounded to three decimal places)
Therefore, the mean (μ) of the x distribution is 72.83 inches and the standard error is approximately 0.712 inches.
(B) Finding p(x > 73.25):
To find the probability of x being greater than 73.25 inches, we need to use the standard normal distribution and convert the given value into a z-score.
1. The z-score, denoted as Z, measures the number of standard deviations a given value is from the mean and can be calculated using the formula: z = (x - μ) / σ.
In this case, x = 73.25 (given height), μ = 72.83 (mean height), and σ = 6.86 (standard deviation).
z = (73.25 - 72.83) / 6.86 ≈ 0.061 (rounded to three decimal places)
2. Using a z-table or statistical software, we can find the area under the standard normal curve to the right of the calculated z-score.
P(x > 73.25) = 1 - P(x ≤ 73.25)
By looking up the z-score of 0.061 in the z-table, we find the corresponding area is approximately 0.524.
P(x > 73.25) = 1 - 0.524 = 0.476 (rounded to three decimal places)
Therefore, the probability of randomly selecting a male college student with a height greater than 73.25 inches is approximately 0.476 or 47.6%.