Question

For a sample of n=64​, the probability of a sample mean being less than 20.5 if u = 21 and sigma = 1.31 is ​(Round to four decimal places as​ needed.) Would the given sample mean be considered​ unusual?

Answer 1

The probability of the sample mean being less than 20.5 is found by calculating the z-score and then looking up this value in a z-table or using statistical software. With a z-score of -3.05, the probability is low, and the sample mean of 20.5 is considered unusual since it exceeds the threshold of two standard deviations away from the population mean.

Step-by-step explanation:

To determine the probability of the sample mean being less than 20.5, we need to use the normal distribution since the sample size is large (n=64). Given the population mean (μ) is 21 and the population standard deviation (σ) is 1.31, we can calculate the z-score for the sample mean of 20.5 using the formula:

Z = (X - μ) / (σ / √n)

Where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Plugging the values into this formula, we get:

Z = (20.5 - 21) / (1.31 / √64) = (20.5 - 21) / (1.31 / 8) = -0.5 / 0.16375 = -3.05

Now we can look up the z-score of -3.05 in a standard normal distribution table or use statistical software to find the corresponding probability. This probability tells us how likely it is to get a sample mean less than 20.5.

Regarding whether the sample mean is considered unusual, a common rule of thumb is that sample means more than two standard deviations away from the population mean are considered unusual. Since a z-score of -3.05 exceeds two standard deviations from the mean, this sample mean would indeed be considered unusual.

Answer 2

The probability of a sample mean less than 20.5 is 0.3513, which is moderately high. Therefore, the given sample mean wouldn't be considered unusual as it falls within a reasonable range based on the population mean and standard deviation.

Step-by-step explanation:

We can calculate the probability using the standard normal distribution and the z-score:

z = (20.5 - 21) / 1.31 ≈ -3.05

Using a z-table or statistical software, the probability (p-value) of a z-score less than -3.05 is approximately 0.3513.

Typically, a p-value below 0.05 indicates a statistically significant deviation from the expected distribution. However, in this case, 0.3513 is not exceptionally low, suggesting the sample mean is not unusual compared to the population.

Therefore, while the sample mean is slightly lower than the population mean, it's not unlikely enough to be considered unusual based on the given information.