Learning Outcomes
- Conduct a hypothesis test on one variance and interpret the conclusion in context
Recall: STANDARD DEVIATION AND VARIANCE
The most common measure of variation, or spread, is the standard deviation. The standard deviation is a number that measures how far data values are from their mean.
To calculate the standard deviation, we need to calculate the variance first. The variance is the average of the squares of the deviations [latex](x- \overline{x})[/latex] values for a sample, or the [latex]x – μ[/latex] values for a population). The symbol [latex]\sigma ^2[/latex] represents the population variance; the population standard deviation [latex]σ[/latex] is the square root of the population variance. The symbol [latex]s^2[/latex] represents the sample variance; the sample standard deviation [latex]s[/latex] is the square root of the sample variance.
The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data.
A test of a single variance assumes that the underlying distribution is normal. The null and alternative hypotheses are stated in terms of the population variance (or population standard deviation). The test statistic is:
[latex]\displaystyle\dfrac{\left(n-1\right)s^2}{\sigma^2}[/latex]
where:
- [latex]n[/latex] = the total number of data
- [latex]s^2[/latex] = sample variance
- [latex]\sigma^2[/latex] = population variance
You may think of [latex]s[/latex] as the random variable in this test. The number of degrees of freedom is [latex]df=n-1[/latex]. A test of a single variance may be right-tailed, left-tailed, or two-tailed. The example below will show you how to set up the null and alternative hypotheses. The null and alternative hypotheses contain statements about the population variance.
Example 1
Math instructors are not only interested in how their students do on exams, on average, but how the exam scores vary. To many instructors, the variance (or standard deviation) may be more important than the average.
Suppose a math instructor believes that the standard deviation for his final exam is five points. One of his best students thinks otherwise. The student claims that the standard deviation is more than five points. If the student were to conduct a hypothesis test, what would the null and alternative hypotheses be?
try it 1
A scuba instructor wants to record the collective depths of each of his students’ dives during their checkout. He is interested in how the depths vary, even though everyone should have been at the same depth. He believes the standard deviation is three feet. His assistant thinks the standard deviation is less than three feet. If the instructor were to conduct a test, what would the null and alternative hypotheses be?
Recall: ORDER OF OPERATIONS
| Please | Excuse | My | Dear | Aunt | Sally |
| parentheses | exponents | multiplication | division | addition | subtraction |
| [latex]( \ )[/latex] | [latex]x^2[/latex] | [latex]\times \ \mathrm{or} \ \div[/latex] | [latex]+ \ \mathrm{or} \ -[/latex] | ||
To calculate the test statistic follow the following steps:
1st find the numerator:
Step 1: Calculate [latex](n-1)[/latex] by reading the problem or counting the total number of data points and then subtract [latex]1[/latex].
Step 2: Calculate [latex]s^2[/latex], and find the variance from the sample. This can be given to you in the problem or can be calculated with the following formula described in Module 2.
[latex]s^2= \frac{\sum (x- \overline{x})^2}{n-1}[/latex]. Note if you are performing a test of a single standard deviation,
Step 3: Multiply the values you got in Step 1 and Step 2.
Note: if you are performing a test of a single standard deviation, in step 2, calculate the standard deviation, [latex]s[/latex], by taking the square root of the variance.
2nd find the denominator: If you are performing a test of a single variance, read the problem or calculate the population variance with the data. If you are performing a test of a single standard deviation, read the problem or calculate the population standard deviation with the data.
Formula for the Population Variance: [latex]\sigma ^2 = \frac{\sum (x- \mu)^2}{N}[/latex]
Formula for the Population Standard Deviation: [latex]\sigma = \sqrt{\frac{\sum (x- \mu)^2}{N}}[/latex]
3rd take the numerator and divide by the denominator.
EXAMPLE 2
With individual lines at its various windows, a post office finds that the standard deviation for normally distributed waiting times for customers on Friday afternoon is 7.2 minutes. The post office experiments with a single, main waiting line and finds that for a random sample of 25 customers, the waiting times for customers have a standard deviation of 3.5 minutes.
With a significance level of 5%, test the claim that a single line causes lower variation among waiting times (shorter waiting times) for customers.
try it 2
The FCC conducts broadband speed tests to measure how much data per second passes between a consumer’s computer and the internet. As of August 2012, the standard deviation of Internet speeds across Internet Service Providers (ISPs) was 12.2 percent. Suppose a sample of 15 ISPs is taken, and the standard deviation is 13.2. An analyst claims that the standard deviation of speeds is more than what was reported. State the null and alternative hypotheses, compute the degrees of freedom and the test statistic, sketch the graph of the p-value, and draw a conclusion. Test at the 1% significance level.
