An important indicator of lung function is forced expiratory volume(FEV), which is the volume of air that a person can expire in one second. Dr. Jones plans to measure FEv in a random sample of n young women from a certain population, and to use the sample mean as an estimate of the population mean. Let E be the event that Jones's sample mean will be within +- 100mLi of the population mean.Assume that the population distribution is normal with mean 3000 mLi and standard deviation 400 mLi. Find Pr{E} if:
A)n=15
mean of sample means = mean of population = 3000
sigma sample means = sigma population/sqrt n
so s sample mean = 400/sqrt(15)
= 103
100/103 = .971 s
so we are interested in between the mean -.971 sigma to the mean +.971 sigma
from tables of normal distribution you can get the probability of being beyond mean +.971 sigma
it is 1-.834 = .166
by symmetry the probability of being below mean -.971 sigma is also .166
so the probability of being above or below the desired range i 2*.166 = .332
so the probability of being within the desired range is
1-.332 = .668 or 67 %
This makes sense because we all know that about 68 % lies within 1 sigma and we are at almost 1 sigma, namely .971 sigma
To find the probability Pr{E} if n=15, we can use the formula for the standard error of the mean, which is calculated as the standard deviation divided by the square root of the sample size.
Given:
Population mean (μ) = 3000 mLi
Standard deviation (σ) = 400 mLi
Sample size (n) = 15
Step 1: Calculate the standard error (SE) of the mean.
SE = σ / √n
SE = 400 / √15
SE ≈ 103.28
Step 2: Determine the range within which the sample mean falls to satisfy the event E.
Based on the event E, the sample mean should be within ±100 mLi of the population mean.
Therefore, the range is 3000 mLi ± 100 mLi, or [2900 mLi, 3100 mLi].
Step 3: Convert the range to z-scores.
To convert the sample mean range to z-scores, we use the formula z = (X - μ) / σ, where X is the mean.
For the lower bound:
z lower = (2900 - 3000) / 400
z lower = -0.25
For the upper bound:
z upper = (3100 - 3000) / 400
z upper = 0.25
Step 4: Calculate the probability Pr{E} using the z-scores.
Pr{E} = P(-0.25 ≤ Z ≤ 0.25)
Using a standard normal distribution table or a calculator, we can find the probability associated with the z-scores.
Pr{E} ≈ P(0.25) - P(-0.25)
Looking up the values in a standard normal distribution table, we find:
P(0.25) ≈ 0.5987
P(-0.25) ≈ 0.4013
Pr{E} ≈ 0.5987 - 0.4013
Pr{E} ≈ 0.1974
Therefore, if n=15, the probability Pr{E} is approximately 0.1974.
To find the probability of event E, we need to calculate the probability that the sample mean falls within +/- 100 mL of the population mean.
Given:
Population mean (μ) = 3000 mL
Standard deviation (σ) = 400 mL
Sample size (n) = 15
Since the sample size is small (n<30) and the population standard deviation is known, we can use the t-distribution to approximate the sampling distribution.
1. Calculate the standard error of the mean:
The standard error of the mean (SE) is calculated as the population standard deviation divided by the square root of the sample size.
SE = σ / sqrt(n)
SE = 400 / sqrt(15)
SE ≈ 103.28 mL
2. Find the t-score for +/- 100 mL:
To determine the t-score, we need to calculate the difference between the upper and lower limits of the range of interest and divide it by the standard error.
t-score = (Upper limit - Lower limit) / SE
t-score = (100 - (-100)) / 103.28
t-score ≈ 1.94
3. Find the probability using the t-distribution:
Using the t-distribution table or a statistical software, we can find the probability associated with the t-score. For df (degrees of freedom) = (n-1) = 14 and t-score = 1.94, we find that the probability associated with this t-score is approximately 0.950.
Therefore, Pr{E} ≈ 0.950, or 95%. There is a 95% probability that Dr. Jones's sample mean will be within +/- 100 mL of the population mean when the sample size is 15.