Suppose that, in fact, the blood cholesterol level of all men aged 20 to 34 follows the normal distribution with mean of u = 188 milligrams per deciliter (mg/dl) and standard deviation = 41 mg/dl.
1. Choose an SRS of 100 men from this population. What is the sampling distribution of x? What is the probability that x takes a value between 185 and 191 mg/dl? This is the probability that x estimates u within ± 3 mg/dl.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z scores.
376.867
To answer this question, we first need to understand the concept of the sampling distribution of the mean. When we take a sample from a population, the distribution of the sample mean is called the sampling distribution of the mean. In this case, we are interested in the sampling distribution of the mean cholesterol level (x) for a sample of 100 men aged 20 to 34.
The sampling distribution of the mean is approximately normally distributed, regardless of the shape of the original population distribution, as long as the sample size is large enough (Central Limit Theorem).
Given that the population follows a normal distribution with a mean of μ = 188 mg/dl and a standard deviation of σ = 41 mg/dl, the sampling distribution of the mean (x) will also follow a normal distribution with the same mean (u = μ = 188 mg/dl) but with a smaller standard deviation (σ_x = σ / sqrt(n)), where n is the sample size.
In our case, the sample size is 100 (n = 100). Therefore, the standard deviation of the sampling distribution will be σ_x = 41 mg/dl / sqrt(100) = 4.1 mg/dl.
Now, we can calculate the probability that x takes a value between 185 and 191 mg/dl. In other words, we need to find the probability that x is within ± 3 mg/dl of the mean.
To do this, we need to convert the values 185 and 191 mg/dl to z-scores. The formula for calculating z-score is:
z = (x - μ) / σ
For 185 mg/dl:
z1 = (185 - 188) / 4.1
For 191 mg/dl:
z2 = (191 - 188) / 4.1
Now, we can use a standard normal distribution table or a calculator to find the probability between z1 and z2. The area under the normal curve between these z-scores represents the probability of x being between 185 and 191 mg/dl.
Alternatively, you can use software such as Excel or statistical calculators to find the probability directly by inputting the mean, standard deviation, and range of values.
Please note that the values obtained from tables or calculators represent the probability for the standardized normal distribution. To convert this back to the original cholesterol values, you can use the formula:
x = μ + zσ
In this case, the probability that x takes a value between 185 and 191 mg/dl is the same as the probability that x estimates u within ± 3 mg/dl.