8.2 A Single Population Mean (Unknown σ)
To find the standard error, we need the population standard deviation, [latex]\sigma[/latex]. Unfortunately, this value isn’t generally unknown. In this situation, the next best thing we can do is to use the sample standard deviation, [latex]s[/latex], as a substitute for [latex]\sigma[/latex]. This substitution works well for large samples but this estimate is off for small sample sizes. In the case of small sample sizes taken from an underlying normal distribution, a different kind of distribution, called a t-distribution gives better results.
Choosing appropriate distribution: What do I use: z or t distribution?
Do we know population standard deviation, σ?
- If YES, use normal distribution (z-distribution)
- If NO, then use t-distribution.
Note: You may use normal distribution if sample size is at least 30 (n ≥ 30) even if σ is unknown. For n ≥ 30, you can use sample standard deviation (s) in place of population standard deviation ([latex]\sigma[/latex]).Note that since the t and z-distributions look similar (see Desmos demo) for larger sample sizes, probability calculations from these distributions will lead to similar results. For this reason, when dealing with large sample sizes from populations with unknown standard deviations (unknown [latex]\sigma[/latex]), many simply prefer to use t-distribution instead of z.
Finding Critical Values
Use Desmos or StatKey. (Note: If you need to use more than 3 decimal places for the critical value, you may want to skip StatKey)
Critical t-value using Desmos | Critical t-value using StatKey (no audio)
Practice
Please complete the following practice exercise:
Finding the critical value t for a desired confidence level
EXAMPLE: MARGIN OF ERROR
A survey of 46 people showed that the respondents spent an average of $31 on their child’s last birthday gift with a standard deviation of $9. Find the critical value for a 95% confidence level, the standard error, and the margin of error. Assume that the population is normally distributed.
HELP ME SOLVE THIS
The mean and the standard deviations given here are about a sample, as it says in the question — a sample of size 46 with a mean of $31 and a standard deviation of $9.
Given facts are:
[latex]n=46[/latex]
[latex]\bar x = $31[/latex]
[latex]s = $9[/latex]. This is not σ (The notation σ represents the population standard deviation. What does s represent?)
Since the population standard deviation is unknown, we use t-distribution. If the sample size is at least 30, the result from using normal distribution is approximately equal to the one from that of using a t-distribution. Therefore, using normal distribution (z-distribution) wouldn’t be too far off, but t-distribution is there, so why not use it.
Critical value: Use a t-distribution to find the critical value. What would be the degrees of freedom for the t-distribution here?
Standard Error: Standard Error is given by [latex]\sigma_{\bar x} = \dfrac{s}{\sqrt n}[/latex]
Now that we have the pieces sorted out, let’s use the EBM (or Margin or Error, ME) formula to find the margin of error.
Margin of Error = (Critical Value) • (Standard Error)
EXAMPLE: MEAN WITH STATISTICS
The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 40.9 for a sample of size 20 and standard deviation 11.7.
Estimate how much the drug will lower a typical patient’s systolic blood pressure (using a 98% confidence level).
Assume the data is from a normally distributed population. Round your answers to 3 decimals.
SHOW SOLUTION
Confidence level, c = 0.98 (for a 98% confidence interval)
Sample info (These include sample statistics):
Sample Mean, [latex]\bar x = 40.9[/latex]
Sample Standard Deviation, [latex]s = 11.7[/latex]
Sample Size, [latex]n = 20[/latex]
Population standard deviation is not known. Sample size is fewer than 30 and the population is normally distributed. Therefore, use a t-distribution.
Note that confidence interval is:
Point Estimate ± Margin of Error
and
Margin of Error = Critical Value • Standard Error, where tc is the critical value and the standard error of the mean = [latex]\dfrac{s}{\sqrt n}[/latex].
Margin of Error = [latex]t_c \cdot \dfrac{s}{\sqrt n}[/latex]
So, confidence interval is:
Point Estimate ± Critical Value • Standard Error
Point Estimate [latex]\pm \:{\color{#e03e2d} {t_c}} \cdot \dfrac{s}{\sqrt n}[/latex].
First, recognize that the sample mean is our point estimate. x̅ = 40.9. And our sample standard deviation is given as well: s = 11.7, while the sample size n = 20.
Let’s update our CI:
Point Estimate ± Margin of Error
= Point Estimate [latex]\pm \:{\color{#e03e2d}{ t_c}} \cdot \dfrac{s}{\sqrt {n}}[/latex]
= [latex]40.9 \pm \:{\color{#e03e2d} {t_c}} \cdot \dfrac{11.7}{\sqrt {20}}[/latex]
Now, we just need the critical value.
With [latex]n = 20[/latex], degrees of freedom, [latex]d.f. = n - 1 = .........[/latex]
Use Desmos or StatKey or TiCalculator to find the critical value, [latex]t_c[/latex], for a 98% confidence level. After you have computed the value, click on Show More below to show critical value and more:
Show More
[latex]t_c= 2.53948319062[/latex]
Let’s plug this [latex]t_c[/latex] into the formula above to find the confidence interval:
[latex]40.9 \pm 2.53948319062[/latex] • [latex]\dfrac{11.7}{\sqrt {20}}[/latex]
= [latex]40.9 \pm 6.64379473907[/latex]
This is our confidence interval in ± notation.
Three Ways to Write Confidence Intervals
1) So, the confidence interval in interval notation:
(34.25620530117019, 47.54379469882981) → Round to 3 decimals: (34.256, 47.544)
2) Confidence interval in tri-inequality notation:
34.25620530117019 < [latex]\mu[/latex] < 47.54379469882981
34.256 < [latex]\mu[/latex] < 47.544
3) Confidence interval in plus-minus notation:
Margin of Error, ME or EBM = 47.54379469882981 − 40.9 = 6.64379469883 ≈ 6.644
Confidence interval: 40.9 ± 6.644
The SUBEDI calculator gives answers in ± notation, whereas the LibreText calculator‘s results are in interval notation. Be sure to convert CI in one notation to another. (See Three Ways to Write a Confidence Interval for additional details on notations).
ONLINE CALCULATOR Approach
SUBEDI Calculator
Go to Confidence Interval for a Mean calculator @ rsubedi.com
Confidence Level (in decimal), [latex]c: \fbox{$\mathstrut \;0.98\;$}[/latex]
Number of Samples
Input Type
Distribution Type to Use?
Sample Size, [latex]n: \fbox{$\mathstrut \;20\;$}[/latex]
Sample Mean, [latex]\bar x: \fbox{$\mathstrut \;40.9\;$}[/latex]
Sample Standard Deviation, [latex]n: \fbox{$\mathstrut \;11.7\;$}[/latex]
CALCULATE
Results show in a panel to the right. CI is displayed in [latex]\pm[/latex] notation.
LibreText Calculator
Go to: Confidence Interval for a Mean With Statistics from the list of online calculators.
Enter the following values and press Calculate.
CALCULATE
Results displayed are:
EXAMPLE: MEAN WITH DATA
You are interested in finding a 90% confidence interval for the mean number of visits for physical therapy patients. The data below show the number of visits for 11 randomly selected physical therapy patients.
| 14 | 12 | 6 | 27 | 12 | 13 | 21 | 20 | 20 | 13 | 19 |
Assume the data is from a normally distributed population. Round your answers to 3 decimals.
SHOW SOLUTION
WHY?
Confidence level, c = 0.90 (for a 90% confidence interval)
Population standard deviation is not known. Sample size is fewer than 30 and the population is normally distributed. Therefore, use a t-distribution.
ONLINE CALCULATOR Approach
SUBEDI Calculator
Go to Confidence Interval for a Mean calculator @ rsubedi.com
Confidence Level (in decimal), [latex]c: \fbox{$\mathstrut \;0.90\;$}[/latex]
Number of Samples
Input Type
Distribution Type to Use?
Enter your data in the spreadsheet column shown for data entry.
Select the above data to copy. Once copied, on SUBEDI Calc click on the first cell of the spreadsheet and paste the data (Control+V or Command + V).
CALCULATE
Results show in a panel to the right. CI is displayed in [latex]\pm[/latex] notation.
LibreText Calculator
Go to: Confidence Interval Calculator with Data from the list of online calculatorsEnter the following values and press Calculate.
Data: (Separate each value with a comma)
[latex]\fbox{$\mathstrut \quad14, 12, 6, 27, 12, 13, 21, 20, 20, 13, 19\quad$}[/latex]
[latex]\text{CL}: \fbox{$\mathstrut \;0.90\;$}[/latex]
CALCULATE
Results displayed are:
