A simple random sample of size n equals=3737 is obtained from a population with μ equals=6969 and σ equals=1515.
(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? Assuming that this condition is true, describe the sampling distribution of x bar
(b) Assuming the normal model can be used, determine
P(x bar < 73.3).
(c) Assuming the normal model can be used, determine P(x bar ≥ 70.3).
(a) In order to use the normal model to compute probabilities involving the sample mean, the distribution of the population must be approximately normal or the sample size must be large enough. If the population distribution is not normal, the Central Limit Theorem states that as long as the sample size is sufficiently large (typically n ≥ 30), the sampling distribution of the sample mean will be approximately normal regardless of the population distribution.
(b) To determine P(x̄ < 73.3), where x̄ represents the sample mean, we need to use the standard normal distribution. The mean of the sample mean (x̄) is equal to the population mean (μ), and the standard deviation of the sample mean (σx̄) is equal to the population standard deviation (σ) divided by the square root of the sample size (√n).
Given:
μ = 6969
σ = 1515
n = 3737
x̄ = 73.3
First, we need to calculate the standard deviation of the sample mean:
σx̄ = σ / √n
= 1515 / √3737
Next, we can standardize the value of x̄ by subtracting the mean and dividing by the standard deviation:
z = (x̄ - μ) / σx̄
= (73.3 - 6969) / (1515 / √3737)
Finally, we can calculate P(x̄ < 73.3) by finding the area under the standard normal distribution curve to the left of the standardized value z using a standard normal distribution table or a statistical software.