# Inferences about One-Pop. Standard Deviation on the TI-83/84

Copyright © 2008 by Stan Brown, Oak Road Systems

Copyright © 2008 by Stan Brown, Oak Road Systems

**Summary:**
This Web page presents a
**downloadable TI-83/84 program** that lets you perform a
hypothesis test on **standard deviation or variance** of a
population, or compute a confidence interval about standard deviation
or variance of a population.

The tests on this page require that
**the underlying population must be normal**.
They are not robust, meaning that even moderate departures from
normality can invalidate your analysis.
See Normality Check on the TI-83/84 or TI-89 for procedures to test whether a
population is normal by testing the sample.

**Outliers are also unacceptable** and must be ruled out.
See Make a Box-Whisker Plot
for an easy way to test for outliers.

**See also:**
Inferences about One Population Standard Deviation gives the statistical
concepts with examples of calculation “by hand” and
in an Excel workbook.

You already know how to test the mean of a population with a t test, or estimate a population mean using a t interval. Why would you want to do that for the standard deviation of a population?

The standard deviation measures **variability**. In many
situations not just the average is important, but
also the variability. For example, suppose you are thinking about
investing in one of two mutual funds. Both show an average annual
growth of 3.8% in the past 20 years, but one has a standard deviation
of 8.6% and the other has a standard deviation of 1.2%. Obviously you
prefer the second one, because with the first one there’s
quite a good chance that you’d have to take a loss if you need
money suddenly.

Industrial processes, too, are monitored not only for average output but for variability within a specified tolerance. If the diameter of ball bearings produced varies too much, many of them won’t fit in their intended application. On the other hand, it costs more money to reduce variability, so you may want to make sure that the variability is not too low either.

This TI-83/84 program can perform hypothesis tests and compute confidence intervals for the standard deviation of a population. Since variance is the square of standard deviation, it can also do those calculations for the variance of a population.

Please observe the cautions above, and
always check the requirements before using this program. If
the requirements are not met, the results from
`SDINFER`

will be, in Gene Wilder’s words from
Young Frankenstein, “doo-doo”.

There are three methods to get the program into your calculator:

- If a classmate has the program on her calculator
(any model TI-83/84), she
can transfer it to yours with the short cable with headphone-style
plugs that comes with all TI
calculators. On your calculator, press [
`2nd`

`x,T,θ,n`

*makes*`LINK`

] [`►`

] [`ENTER`

], and then on hers press [`2nd`

`x,T,θ,n`

*makes*`LINK`

] [`3`

], select the program, then press [`►`

] [`ENTER`

]. - Or, download SDINFER.ZIP (22 KB, updated Aug 9, 2008), unzip it, and transfer file SDINFER.8XP to your calculator. (This requires TI-Connect or TI-Graph Link software software and a cable.)
- Or, as a last resort, key in the program. See SDINFER.PDF and SDINFER_HINTS.HTM in the SDINFER.ZIP file.

To use the program, first check the requirements for your
sample; see Cautions above.
Then press [`PRGM`

], select
`SDINFER`

, and press [`ENTER`

] [`ENTER`

].
When prompted, enter the standard deviation and size of the sample,
pressing [`ENTER`

] after each one. If you know the variance of
the sample rather than the standard deviation, use the square root
operation since s is the square root of the variance s².

The program then presents you with a five-item menu: confidence interval for the population standard deviation σ, confidence interval for the population variance σ², and three hypothesis tests for σ or σ² less than, different from, or greater than a number. Make your selection by pressing the appropriate number.

If you select one of the confidence intervals, the program will prompt you for the confidence level and then compute the interval. Because this involves a process of successive approximations, it can take some time, so please be patient.

The program displays the endpoints of the interval on screen
and also leaves them in variables `L`

and `H`

in
case you want to use them in further calculations. You can include
them in any formula by [`ALPHA`

`)`

*makes* `L`

] and [`ALPHA`

`^`

*makes* `H`

].

If you select one of the hypothesis tests, the program will
prompt you for σ, the population standard deviation in the
null hypothesis. If your H_{0} is about population variance
σ² rather than σ, use the square root symbol to
convert the hypothetical variance to standard deviation.

The program then displays the χ²_{o} test statistic, the
degrees of freedom, and the p-value. These are also left in variables
`X`

, `D`

, and `P`

in case you wish to use
them in further calculations. You can include them in any formula with
[x,T,θ,n], [`ALPHA`

`x`

^{-1 makes}` `

`D`

], and [`ALPHA`

`8`

*makes* `P`

].

```
```## Examples

**Example 1**: A machine packs cereal into
boxes, and you require a standard deviation of no more than five
grams. You randomly select and weigh 45 boxes and find a sample
standard deviation of 6.2 grams. Is the machine operating
within specification?

You have tested the sample and find that it
is normally distributed with no outliers, so you are confident that
the population is also normally distributed.

**Solution**: n = 45, s = 6.2,
σ_{o} = 5. Your hypotheses are

H_{0}: σ ≤ 5, the machine is within spec
(some books would say H_{0}: σ=5)

H_{1}: σ > 5, the machine is not working right

No α was specified, but for an industrial process with no
possibility of human injury α = 0.05 seems
appropriate.

Run the `SDINFER`

program. Enter s:6.2 and
n:45, and select `5:TEST σ>CONST`

. Enter 5
for H_{0} σ.

The results are shown at right. The test statistic is
χ²_{o} = 67.65 with 44 degrees of freedom, and the
p-value is 0.0125.

Since p<α, you reject H_{0} and accept H_{1}.
At the 0.05 level of significance, the population standard deviation
σ is greater than 5, and the machine is not operating within
specificaton.

**Example 2**:
You have a random sample of size 20, with a standard deviation of 125. You
have good reason to believe that the underlying population is normal.
Is the population standard deviation different from 100, at the 0.05
significance level?

**Solution**:
n = 20, s = 125, σ_{o} = 100,
α = 0.05. Your hypotheses are

H_{0}: σ = 100

H_{1}: σ ≠ 100

This time in the `SDINFER`

program you select
`4:TEST σ≠CONST`

.

Results are shown at right. χ²_{o} = 29.69 with
19 degrees of freedom, and the p-value is 0.1118.

p>0; fail to reject H_{0}. At the 0.05 significance level,
you can’t say whether the population standard deviation
σ is different from 100 or not.

**Example 3**: Of several thousand students
who took the same exam, 40 papers were selected randomly and
statistics were computed. The standard deviation of the sample was 17
points. Estimate the standard deviation of the population, with 95%
confidence. (Recall that test scores are normally distributed.)

**Solution**:
Run `SDINFER`

and select
`1:σ INTERVAL`

with a C-level of 95 or .95.
The results screen is shown at right.

Conclusion: We’re 95% confident that the standard
deviation of test scores for all students is between 13.9 and
21.8.

**Example 4**:
Heights of U.S. males aged 18–25 are normally distributed. You
take a random sample of 100 from that population and find a mean of
65.3 in and a variance of 7.3 in². (Remember that
the units of variance are the square of the units of the original
measurement.)

Estimate the mean and variance of the height of U.S. males
aged 18–25, with 95% confidence.

**Solution for mean**:
Computing a confidence interval for the mean is a straightforward
`TInterval`

. Just remember that for `Sx`

the
calculator wants the sample standard deviation, but you have the
sample variance, which is s². Therefore you take the square root
of sample variance to get sample standard deviation, as shown in the
input screen at near right.

The output screen at far right shows the confidence interval.
We’re 95% confident that the mean height of U.S. males aged
18–25 is between 64.8 and 65.8 in.

**Solution for variance**:
Run the `SDINFER`

program. Enter s:√7.3 and n:100.
Select `2:σ² INTERVAL`

and enter C-Level:.95 (or
95). The program computes the confidence interval for population
variance as 5.6 ≤ σ² ≤ 9.9.
We’re 95% confident that the variance in heights of U.S. males
aged 18–25 is between 5.6 and 9.9 in².

**Complete answer**:
We’re 95% confident that the heights of U.S. males aged
18–25 have mean 64.8–65.9 in and variance
5.6–9.9 in².

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