Confidence Interval for mean sigma unknown examples

Confidence Interval-sigma unknown

This tutorial covers examples on confidence interval for the population mean when the population standard deviation is unknown.

Example 1

A new brand of laptop battery is produced by a company. The company claims that the battery will last for an extended period of time before a recharge is necessary. A sample of 40 batteries is tested for the length of usage time to recharge. The sample results are as follow:

Sample size: 40, Sample Mean: 6.5 hrs, Sample Standard Deviation: 1.3 hrs.

Construct a 99 % confidence interval for the average length of usage time to recharge

Solution

Given that sample size $n = 40$, sample mean $\overline{X}= 6.5$, sample standard deviation $s = 1.3$.
The confidence level is $1-\alpha = 0.99$.

Step 1 Specify the confidence level $(1-\alpha)$

Confidence level is $1-\alpha = 0.99$. Thus, the level of significance is $\alpha = 0.01$.

Step 2 Given information

Given that sample size $n =40$, sample mean $\overline{X}=6.5$, sample standard deviation $s=1.3$.

Step 3 Specify the formula

$100(1-\alpha)$% confidence interval for the population mean $\mu$ is
$$ \begin{aligned} \overline{X} - E \leq \mu \leq \overline{X} + E \end{aligned} $$

where $E = t_{(\alpha/2,n-1)} \frac{s}{\sqrt{n}}$, and $t_{\alpha/2, n-1}$ is the $t$ value providing an area of $\alpha/2$ in the upper tail of the students' $t$ distribution.

Step 4 Determine the critical value

The critical value of $t$ for given level of significance and $n-1$ degrees of freedom is $t_{\alpha/2,n-1}$.

t-critical value for CI
t-critical value for CI

Thus $t_{\alpha/2,n-1} = t_{0.005,40-1}= 2.708$.

Step 5 Compute the margin of error

The margin of error for mean is
$$ \begin{aligned} E & = t_{(\alpha/2,n-1)} \frac{s}{\sqrt{n}}\\ & = 2.708 \frac{1.3}{\sqrt{40}} \\ & = 0.557. \end{aligned} $$

Step 6 Determine the confidence interval

$99$% confidence interval estimate for population mean is

$$ \begin{aligned} \overline{X} - E & \leq \mu \leq \overline{X} + E\\ 6.5 - 0.557 & \leq \mu \leq 6.5 + 0.557\\ 5.943 &\leq \mu \leq 7.057. \end{aligned} $$

Thus, $99$% confidence interval estimate for population mean is $(5.943,7.057)$.

Interpretation

We can be $99$% confident that the average length of usage time to recharge is between $5.943$ and $7.057$.

Example 2

Find the 95% confidence interval for the mean number of ounces of coffee that a machine dispenses in 12 ounce cups. Assume the variable is normally distributed. The data is shown below:

12.03, 12.10, 12.02, 11.98, 12.00, 12.05, 11.97.

Solution

$x_i$ $x_i^2$
12.03 144.72
12.10 146.41
12.02 144.48
11.98 143.52
12.00 144.00
12.05 145.20
11.97 143.28
Total 84.15 1011.62

Sample mean

The sample mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{84.15}{7}\\ &=12.0214\text{ ounce} \end{aligned} $$

The average of ounces of coffee is $12.0214$ ounce.

Sample variance

Sample variance of $X$ is

$$ \begin{aligned} s_x^2 &=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\frac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)\\ &=\dfrac{1}{6}\bigg(1011.6151-\frac{(84.15)^2}{7}\bigg)\\ &=\dfrac{1}{6}\big(1011.6151-\frac{7081.2225}{7}\big)\\ &=\dfrac{1}{6}\big(1011.6151-1011.60321\big)\\ &= \frac{0.01189}{6}\\ &=0.002 \end{aligned} $$

Sample standard deviation

The sample standard deviation is

$$ \begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{0.002}\\ &=0.0445 \text{ ounce} \end{aligned} $$

Sample size $n = 20$, sample mean $\overline{X}= 12.0214$, sample standard deviation $s = 0.0445$.

The confidence level is $1-\alpha = 0.95$.

Step 1 Specify the confidence level $(1-\alpha)$

Confidence level is $1-\alpha = 0.95$. Thus, the level of significance is $\alpha = 0.05$.

Step 2 Given information

Sample size $n =20$, sample mean $\overline{X}=12.0214$, sample standard deviation $s=0.0445$.

Step 3 Specify the formula

$100(1-\alpha)$% confidence interval for the population mean $\mu$ is

$$ \begin{aligned} \overline{X} - E \leq \mu \leq \overline{X} + E \end{aligned} $$

where $E = t_{(\alpha/2,n-1)} \frac{s}{\sqrt{n}}$, andand $t_{\alpha/2, n-1}$ is the $t$ value providing an area of $\alpha/2$ in the upper tail of the students' $t$ distribution.

Step 4 Determine the critical value

The critical value of $t$ for given level of significance and $n-1$ degrees of freedom is $t_{\alpha/2,n-1}$.

t-critical value for CI
t-critical value for CI

Thus $t_{\alpha/2,n-1} = t_{0.025,20-1}= 2.093$.

Step 5 Compute the margin of error

The margin of error for mean is
$$ \begin{aligned} E & = t_{(\alpha/2,n-1)} \frac{s}{\sqrt{n}}\\ & = 2.093 \frac{0.0445}{\sqrt{20}} \\ & = 0.0208. \end{aligned} $$

Step 6 Determine the confidence interval

$95$% confidence interval estimate for population mean is

$$ \begin{aligned} \overline{X} - E & \leq \mu \leq \overline{X} + E\\ 12.0214 - 0.021 & \leq \mu \leq 12.0214 + 0.021\\ 12.0006 &\leq \mu \leq 12.0422. \end{aligned} $$

Thus, $95$% confidence interval estimate for population mean is $(12.0006,12.0422)$.

Interpretation

We can be $95$% confident that the mean number of ounces of coffee that a machine dispenses in 12 ounce cups is between $12.0006$ and $12.0422$.

Reference

You can read step by step tutorial on Confidence Interval for mean sigma unknown, tutorial will help you to understand how to construct confidence interval for population mean when the population standard deviation is unknown.

You can also use calculator to compute the confidence interval for population mean when the population standard deviation is unknown.

Confidence Interval for mean sigma unknown

Confidence Interval for Mean (t) Calculator

Let me know in the comments if you have any questions on confidence interval for mean for sigma unknown with examples and your thought on this article.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.

Leave a Comment