Questions: The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual. For a sample of n=70, find the probability of a sample mean being less than 21.1 if μ=21 and σ=1.27. For a sample of n=70, the probability of a sample mean being less than 21.1 if μ=21 and σ=1.27 is (Round to four decimal places as needed.)

The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual.

For a sample of n=70, find the probability of a sample mean being less than 21.1 if μ=21 and σ=1.27.

For a sample of n=70, the probability of a sample mean being less than 21.1 if μ=21 and σ=1.27 is
(Round to four decimal places as needed.)

Transcript text: The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual. For a sample of $n=70$, find the probability of a sample mean being less than 21.1 if $\mu=21$ and $\sigma=1.27$. For a sample of $n=70$, the probability of a sample mean being less than 21.1 if $\mu=21$ and $\sigma=1.27$ is $\square$ (Round to four decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate the Z-score

To find the probability of a sample mean being less than $ 21.1 $, we first calculate the Z-score for the sample mean using the formula:

\[ Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \]

Where:

$ \bar{x} = 21.1 $ (sample mean)
$ \mu = 21 $ (population mean)
$ \sigma = 1.27 $ (population standard deviation)
$ n = 70 $ (sample size)

Calculating the standard error:

\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{1.27}{\sqrt{70}} \approx 0.151 \]

Now substituting into the Z-score formula:

\[ Z = \frac{21.1 - 21}{0.151} \approx 0.6588 \]

Step 2: Find the Probability

Using the Z-score, we can find the probability that the sample mean is less than $ 21.1 $:

\[ P(Z < 0.6588) = \Phi(0.6588) \]

From the standard normal distribution table, we find:

\[ P(Z < 0.6588) \approx 0.745 \]

Step 3: Determine if the Sample Mean is Unusual

A sample mean is typically considered unusual if the probability of obtaining such a sample mean is less than $ 0.05 $ (or $ 5\% $). Since $ P \approx 0.745 $, which is greater than $ 0.05 $, we conclude that the sample mean of $ 21.1 $ is not unusual.