Thank you for visiting Use technology to solve the following problem A sample of size 45 will be drawn from a population with a mean of 94 and a. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
We start by noting that the sampling distribution of the sample mean, [tex]$\bar{x}$[/tex], is approximately normal with mean
[tex]$$
\mu_{\bar{x}} = 94,
$$[/tex]
and standard error
[tex]$$
\sigma_{\bar{x}} = \frac{14}{\sqrt{45}}.
$$[/tex]
Step 1. Calculate the Standard Error:
The standard error is given by
[tex]$$
\sigma_{\bar{x}} = \frac{14}{\sqrt{45}} \approx 2.087.
$$[/tex]
Step 2. Find the Z-score for the 32nd Percentile:
For a standard normal distribution, find the [tex]$z$[/tex]-value such that
[tex]$$
P(Z \le z) = 0.32.
$$[/tex]
Looking up or using a calculator for the standard normal distribution, we get
[tex]$$
z \approx -0.4677.
$$[/tex]
Step 3. Calculate the 32nd Percentile of [tex]$\bar{x}$[/tex]:
The corresponding value of [tex]$\bar{x}$[/tex] is found by
[tex]$$
\bar{x} = \mu_{\bar{x}} + z \cdot \sigma_{\bar{x}}.
$$[/tex]
Substitute the values:
[tex]$$
\bar{x} = 94 + (-0.4677)(2.087) \approx 94 - 0.977 \approx 93.023.
$$[/tex]
Rounded to one decimal place, the 32nd percentile of [tex]$\bar{x}$[/tex] is approximately
[tex]$$
93.0.
$$[/tex]
Thus, the answer is [tex]$\boxed{93.0}$[/tex].
[tex]$$
\mu_{\bar{x}} = 94,
$$[/tex]
and standard error
[tex]$$
\sigma_{\bar{x}} = \frac{14}{\sqrt{45}}.
$$[/tex]
Step 1. Calculate the Standard Error:
The standard error is given by
[tex]$$
\sigma_{\bar{x}} = \frac{14}{\sqrt{45}} \approx 2.087.
$$[/tex]
Step 2. Find the Z-score for the 32nd Percentile:
For a standard normal distribution, find the [tex]$z$[/tex]-value such that
[tex]$$
P(Z \le z) = 0.32.
$$[/tex]
Looking up or using a calculator for the standard normal distribution, we get
[tex]$$
z \approx -0.4677.
$$[/tex]
Step 3. Calculate the 32nd Percentile of [tex]$\bar{x}$[/tex]:
The corresponding value of [tex]$\bar{x}$[/tex] is found by
[tex]$$
\bar{x} = \mu_{\bar{x}} + z \cdot \sigma_{\bar{x}}.
$$[/tex]
Substitute the values:
[tex]$$
\bar{x} = 94 + (-0.4677)(2.087) \approx 94 - 0.977 \approx 93.023.
$$[/tex]
Rounded to one decimal place, the 32nd percentile of [tex]$\bar{x}$[/tex] is approximately
[tex]$$
93.0.
$$[/tex]
Thus, the answer is [tex]$\boxed{93.0}$[/tex].
Thank you for reading the article Use technology to solve the following problem A sample of size 45 will be drawn from a population with a mean of 94 and a. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Jeany
