A large pizzeria makes 1000 pies per day. The mean diameter of u=10.1 inches and the standard deviation is 0.5 inches. A random sample of 3
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
![### Statistics Practice Problem
#### Problem Statement
A large pizzeria makes 1000 pies per day. The mean diameter of these pies is \(\mu = 10.1\) inches, and the standard deviation is 0.5 inches. A random sample of 36 pies is analyzed.
Find:
a) \(\mu_{\bar{x}}\)
b) The standard error of the mean.
c) \(P(\bar{X} \leq 9.8)\)
d) \(P(X \leq 9.8)\)
#### Solution Approach
a) **\(\mu_{\bar{x}}\)**
The mean of the sample mean \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\). Therefore:
\[ \mu_{\bar{x}} = \mu = 10.1 \text{ inches} \]
b) **The standard error of the mean**
The standard error of the mean (SEM) is calculated using the formula:
\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]
where \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
\[ \sigma_{\bar{x}} = \frac{0.5}{\sqrt{36}} = \frac{0.5}{6} = 0.0833 \text{ inches} \]
c) **\(P(\bar{X} \leq 9.8)\)**
To find this probability, we need to use the Z-score formula for the sample mean:
\[ Z = \frac{\bar{X} - \mu_{\bar{x}}}{\sigma_{\bar{x}}} \]
\[ Z = \frac{9.8 - 10.1}{0.0833} \approx -3.6 \]
Using the Z-table to find the probability:
\[ P(\bar{X} \leq 9.8) \approx P(Z \leq -3.6) \approx 0.0002 \]
d) **\(P(X \leq 9.8)\)**
To find this probability for a single pie with diameter \(X\):
\[ Z = \frac{X - \mu}{\sigma} \]
\[ Z = \frac{9.8 - 10.1}{0.5} = -0.6 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc148dc64-5945-4b64-a9dd-49f10451afec%2F2303b0ea-f86f-4dcc-8be2-2798fc2fdc16%2Fhps3vt8_processed.jpeg&w=3840&q=75)
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