A population is normally distributed with a man of 100 and a standard deviation of 15. is it unusual for themean of a sample of 3 to be 115 or more? Why or why not?
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff. Essentially it becomes just SEm.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to that Z score.
Use that to make you decision.
To determine if it is unusual for the mean of a sample of 3 to be 115 or more, we can calculate the sample mean and compare it with the population mean.
To calculate the mean of a sample, we sum up all the values in the sample and divide it by the sample size. In this case, since the sample size is 3, we need to take the average of the three values.
However, before calculating the sample mean, we must remember that the distribution of sample means is also normally distributed, but with its own mean and standard deviation. The mean of the sample means, also known as the expected value, is equal to the population mean. The standard deviation of the sample means, also known as the standard error, is equal to the population standard deviation divided by the square root of the sample size (n).
Given the population mean (μ) of 100, the standard deviation (σ) of 15, and the sample size (n) of 3, we can calculate the standard error:
Standard Error (SE) = σ / √n
SE = 15 / √3 ≈ 8.66
Since the distribution of the sample means is normally distributed, we can use the Z-score formula to standardize the sample mean:
Z-score = (x - μ) / SE
Here, x is the sample mean we want to evaluate (115), μ is the population mean (100), and SE is the standard error (8.66).
Z-score = (115 - 100) / 8.66 ≈ 1.73
Now we need to determine if the Z-score is considered unusual or not. We can do this by looking up the Z-score in the standard normal distribution table or by using statistical software. If the Z-score corresponds to a very small p-value (typically below 0.05), then it is considered unusual.
According to the standard normal distribution table, a Z-score of 1.73 corresponds to a p-value of approximately 0.0427. This means that the probability of observing a sample mean of 115 or more, assuming the population mean is 100, is approximately 0.0427.
Since the p-value is less than the typical significance level of 0.05, we can conclude that a sample mean of 115 or higher is considered unusual.