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.
2. Choose an SRS of 1000 men from this population. Now what is the probability that x falls within ± 3 mg/dl of u? The larger sample is much more likely to give an accurate estimate of u.
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.
1. The sampling distribution of x, the sample mean cholesterol level, is also a normal distribution. The mean of the sampling distribution is equal to the population mean, u, which is 188 mg/dl. The standard deviation of the sampling distribution, also known as the standard error, is equal to the population standard deviation divided by the square root of the sample size. In this case, the standard deviation of the sampling distribution is 41 mg/dl divided by the square root of 100, which is 4.1 mg/dl.
To calculate the probability that x takes a value between 185 and 191 mg/dl, we can convert these values to z-scores using the formula z = (x - u) / (standard deviation). For 185 mg/dl:
z = (185 - 188) / 4.1 ≈ -0.73
For 191 mg/dl:
z = (191 - 188) / 4.1 ≈ 0.73
We can then find the probability by looking up these z-scores in a standard normal distribution table or by using a statistical calculator. The probability that x takes a value between 185 and 191 mg/dl is the same as the probability that z falls between -0.73 and 0.73.
P(-0.73 < z < 0.73) ≈ 0.579
Therefore, the probability that x estimates u within ± 3 mg/dl is approximately 0.579.
2. When we choose an SRS of 1000 men from the population, the standard deviation of the sampling distribution decreases. The standard deviation of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size. In this case, the standard deviation of the sampling distribution is 41 mg/dl divided by the square root of 1000, which is 1.3 mg/dl.
To calculate the probability that x falls within ± 3 mg/dl of u, we can follow a similar process as in the previous question. Convert the values to z-scores and then find the probability.
For 185 mg/dl:
z = (185 - 188) / 1.3 ≈ -2.31
For 191 mg/dl:
z = (191 - 188) / 1.3 ≈ 2.31
The probability that x falls within ± 3 mg/dl of u is the same as the probability that z falls between -2.31 and 2.31.
P(-2.31 < z < 2.31) ≈ 0.989
Therefore, when we choose an SRS of 1000 men from the population, the probability that x falls within ± 3 mg/dl of u is approximately 0.989. This larger sample size gives a more accurate estimate of u.
To answer these questions, we need to understand the concept of sampling distribution and how it relates to estimating the population mean.
1. The sampling distribution of x:
When we take a random sample of size n from a population, the sampling distribution of x is the distribution of all possible sample means (x̄) that could be obtained from samples of size n. The sampling distribution of x follows the same shape as the population distribution but has a smaller standard deviation.
In this case, since we are sampling 100 men from the population, the sampling distribution of x will also follow a normal distribution with the same mean u = 188 mg/dl, but with a smaller standard deviation. The standard deviation of the sampling distribution of x (also known as the standard error) is given by σ/√n, where σ is the population standard deviation and n is the sample size.
2. Probability that x takes a value between 185 and 191 mg/dl:
To calculate this probability, we need to compute the z-scores for the upper and lower limits of the interval and refer to the standard normal distribution.
The z-score for a particular value of x can be calculated using the formula:
z = (x - u) / (σ / √n)
For the lower limit of 185 mg/dl:
z_lower = (185 - 188) / (41 / √100)
For the upper limit of 191 mg/dl:
z_upper = (191 - 188) / (41 / √100)
Once we have the z-scores, we can look up the corresponding probabilities from the standard normal distribution table or use technology such as calculators or statistical software.
2. Probability that x falls within ± 3 mg/dl of u for a sample size of 1000:
Since we are now sampling 1000 men, the standard error of the sampling distribution of x decreases because of the larger sample size. The standard deviation of the sampling distribution of x is now σ / √1000.
To calculate the probability that x falls within ± 3 mg/dl of u, we use the same approach as before, but with the updated standard error.
z_lower = (185 - 188) / (41 / √1000)
z_upper = (191 - 188) / (41 / √1000)
Again, we can refer to the standard normal distribution table or use technology to find the probabilities associated with these z-scores.
In summary, the larger sample size of 1000 men in the second question leads to a smaller standard error and, consequently, a higher probability of obtaining an accurate estimate of the population mean within ± 3 mg/dl compared to the smaller sample size of 100 men in the first question.