Given the sample size 400 with the sample mean 99, the population mean 100 and computed value of z statistic at 2.5, the value of population standard deviation will be
8
Calculate Population Standard Deviation from Z-Statistic
This problem requires us to calculate the population standard deviation (\(\sigma\)) using the provided information: sample size, sample mean, population mean, and the computed z-statistic.
Understanding the Z-Statistic Formula
The z-statistic is a measure used in hypothesis testing to determine how many standard errors the sample mean is away from the population mean. When the population standard deviation is known, the formula for the z-statistic is:
\[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]Where:
- \( z \) is the computed z-statistic
- \( \bar{x} \) is the sample mean
- \( \mu \) is the population mean
- \( \sigma \) is the population standard deviation
- \( n \) is the sample size
Given Information
From the question, we have the following values:
- Sample size, \( n = 400 \)
- Sample mean, \( \bar{x} = 99 \)
- Population mean, \( \mu = 100 \)
- Computed z-statistic, \( z = 2.5 \)
Step-by-Step Calculation of Population Standard Deviation
We need to rearrange the z-statistic formula to solve for the population standard deviation (\(\sigma\)).
The formula is:
\[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]Substitute the given values into the formula:
\[ 2.5 = \frac{99 - 100}{\frac{\sigma}{\sqrt{400}}} \]Simplify the numerator and the square root in the denominator:
\[ 2.5 = \frac{-1}{\frac{\sigma}{20}} \]Now, rearrange the equation to solve for \( \sigma \). Multiply both sides by \( \frac{\sigma}{20} \):
\[ 2.5 \times \left(\frac{\sigma}{20}\right) = -1 \]Multiply 2.5 by \( \sigma/20 \):
\[ \frac{2.5 \sigma}{20} = -1 \]Multiply both sides by 20:
\[ 2.5 \sigma = -1 \times 20 \] \[ 2.5 \sigma = -20 \]Divide both sides by 2.5:
\[ \sigma = \frac{-20}{2.5} \] \[ \sigma = -8 \]However, the standard deviation cannot be a negative value. The z-statistic value of 2.5 likely represents the magnitude of the difference in terms of standard errors, i.e., \( |z| = 2.5 \). Let's consider the absolute difference between the sample mean and population mean: \( |\bar{x} - \mu| = |99 - 100| = |-1| = 1 \).
Using the magnitude in the calculation:
\[ |z| = \frac{|\bar{x} - \mu|}{\frac{\sigma}{\sqrt{n}}} \] \[ 2.5 = \frac{|99 - 100|}{\frac{\sigma}{\sqrt{400}}} \] \[ 2.5 = \frac{|-1|}{\frac{\sigma}{20}} \] \[ 2.5 = \frac{1}{\frac{\sigma}{20}} \] \[ 2.5 = \frac{1 \times 20}{\sigma} \] \[ 2.5 = \frac{20}{\sigma} \]Now, solve for \( \sigma \):
\[ \sigma = \frac{20}{2.5} \] \[ \sigma = 8 \]Result
The calculated value for the population standard deviation is 8.
Matching with Options
Let's compare our result with the given options:
- Option 1: 59.17
- Option 2: 123.46
- Option 3: 8
- Option 4: Cannot be determined
Our calculated value matches Option 3.
| Symbol | Description | Value |
|---|---|---|
| \(n\) | Sample Size | 400 |
| \(\bar{x}\) | Sample Mean | 99 |
| \(\mu\) | Population Mean | 100 |
| \(z\) | Computed Z-statistic | 2.5 |
| \(\sigma\) | Population Standard Deviation (Calculated) | 8 |
Revision Table: Key Statistical Concepts
| Concept | Formula (Relevant) | Description |
|---|---|---|
| Z-statistic | \( z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \) | Measures how many standard errors a sample mean is from the population mean. |
| Standard Deviation (\(\sigma\)) | N/A (Calculated Here) | A measure of the dispersion or spread of data in the population. |
| Sample Mean (\(\bar{x}\)) | \( \bar{x} = \frac{\sum x_i}{n} \) | The average of the values in a sample. |
| Population Mean (\(\mu\)) | \( \mu = \frac{\sum x_i}{N} \) | The average of all values in the entire population. |
| Sample Size (\(n\)) | N/A | The number of observations in a sample. |
Additional Information: Z-Tests and Standard Error
The z-statistic is commonly used in a z-test, which is a type of hypothesis test. A z-test is appropriate when the sample size is large (typically n > 30) or when the population standard deviation is known. In this problem, we have a large sample size (n=400), which makes the use of the z-statistic appropriate.
The term \( \frac{\sigma}{\sqrt{n}} \) in the denominator of the z-statistic formula is known as the standard error of the mean. It represents the standard deviation of the sampling distribution of the sample means. It tells us how much the sample mean is expected to vary from the population mean due to random sampling.
In this specific calculation, we effectively used the formula \( \sigma = \frac{|\bar{x} - \mu| \times \sqrt{n}}{|z|} \) derived from the z-statistic formula to find the population standard deviation.
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