To estimate the
mean score � of those who took the Medical College Admission
Test on your campus, you will obtain the scores of an
SRS of students. From published information you know that
the scores are approximately Normal with standard deviation
about 6.4. You want your sample mean xbar to estimate mu � with
an error of no more than 1 point in either direction.
(a) What standard deviation must x have so that 99.7% of all
samples give an xbar within 1 point of mu �? (Use the 68–95–99.7
rule.)
(b) How large an SRS do you need in order to reduce the
standard deviation of xbar to the value you found in part (a)?
Must have 0.3333
m,kj
(a) To estimate the standard deviation of x, we can use the 68–95–99.7 rule, also known as the empirical rule. According to this rule, 99.7% of the data falls within 3 standard deviations of the mean.
Since we want xbar to be within 1 point of μ, we can consider this as 1 standard deviation. Therefore, we need to find the standard deviation x such that 99.7% of xbar values fall within 1 point of μ.
Using the empirical rule, we know that 99.7% corresponds to 3 standard deviations. Thus, we can set up the following equation:
3x = 1
Solving for x, we get:
x = 1/3
Therefore, the standard deviation x must be 1/3.
(b) To reduce the standard deviation of xbar to the value found in part (a), we need to determine the sample size (n).
The formula for the standard deviation of xbar (σxbar) is given by:
σxbar = σ / √n
We want σxbar to be equal to 1/3 (found in part (a)). Therefore, we can set up the equation:
1/3 = 6.4 / √n
Solving for n:
√n = (6.4 * 3) / 1
√n = 19.2
n = (19.2)^2 = 368.64
Therefore, we need a sample size of at least 369 students in order to reduce the standard deviation of xbar to the desired value.
To estimate the mean score (μ) of those who took the Medical College Admission Test, we will obtain the scores of a Simple Random Sample (SRS) of students. From published information, we know that the scores are approximately normally distributed with a standard deviation of 6.4.
(a) To estimate μ with an error of no more than 1 point in either direction, we want our sample mean (x̄) to fall within 1 point of μ. Using the 68–95–99.7 rule, we know that approximately 99.7% of all samples will fall within 3 standard deviations of the population mean.
Therefore, we need to find the standard deviation (σ) of x, so that 99.7% of all samples give an x̄ within 1 point of μ. Since 3 standard deviations cover 99.7% of the samples, we set up the following equation:
3σ = 1
Now we solve for σ:
σ = 1/3
So, the standard deviation of x must be 1/3.
(b) To reduce the standard deviation of x̄ to the value found in part (a), we need to determine the required sample size.
The standard deviation of x̄, also known as the standard error (SE) of the mean, is given by σ/√n, where n is the sample size. We want σ/√n to be equal to 1/3.
Therefore, we set up the equation:
1/3 = 6.4/√n
Now we solve for n:
√n = 6.4/(1/3)
√n = 6.4 * 3
√n = 19.2
n = (19.2)^2
n ≈ 369.64
So, we need a sample size of at least 370 in order to reduce the standard deviation of x̄ to the value found in part (a).