The population mean and standard deviation are given. Find the required probability and determine whether the given sample mean would be considered unusual.
For a sample of n=36, find the probability of a sample mean being less than 12.2 if M=12 and O=0.95
0.8962
To solve this problem, we can use the concept of the sampling distribution of the sample mean. The mean of the sampling distribution is equal to the population mean, and the standard deviation of the sampling distribution, also known as the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size.
First, we need to calculate the standard error of the mean (SEM) using the formula SEM = σ / √n, where σ represents the population standard deviation and n represents the sample size. In this case, σ = 0.95 and n = 36, so we have SEM = 0.95 / √36 = 0.95 / 6 = 0.1583.
Next, we can use the standard normal distribution to find the probability. Since the sample mean follows an approximately normal distribution (by the Central Limit Theorem), we can convert the sample mean to a z-score using the formula z = (x - μ) / SEM, where x represents the sample mean and μ represents the population mean.
In this case, x = 12.2 and μ = 12, so the z-score is z = (12.2 - 12) / 0.1583 = 0.632.
To find the probability of the sample mean being less than 12.2, we need to find the area under the standard normal curve to the left of the z-score. This can be done by looking up the z-score in a standard normal distribution table or by using a calculator or statistical software.
For example, using a standard normal distribution table, we can find that the area to the left of z = 0.632 is approximately 0.7357.
Therefore, the probability of a sample mean being less than 12.2 is 0.7357, or 73.57%.
To determine whether the given sample mean would be considered unusual, we can compare it to the confidence interval. If the sample mean falls outside the confidence interval, it would be considered unusual. A commonly used confidence level is 95%.
To create the confidence interval for the sample mean, we can use the formula:
Confidence Interval = (sample mean - critical value * SEM, sample mean + critical value * SEM)
Since the critical value for a 95% confidence interval is approximately ±1.96 (assuming a large sample size), we can calculate the confidence interval as follows:
Confidence Interval = (12 - 1.96 * 0.1583, 12 + 1.96 * 0.1583)
= (11.6873, 12.3127)
If the given sample mean of 12.2 falls within this confidence interval, it would not be considered unusual. However, if it falls outside this interval, it would be considered unusual.