Two Variances - Graphing Calculator: Videos & Practice Problems

Performing a two-sample F-test for variance involves comparing standard deviations to test hypotheses about population variances. Using a graphing calculator's two-sample F-test function simplifies calculating the F-statistic and p-value, essential for determining if one variance is significantly greater than another at a given alpha level (e.g., 0.05). When given sample statistics or raw data, inputting values correctly and selecting the appropriate one-sided alternative hypothesis enables efficient hypothesis testing. Rejecting the null hypothesis occurs if the p-value is less than alpha, indicating evidence that one population variance exceeds the other, crucial for statistical inference and decision-making.

Hypothesis Test for Two Variances Using a TI-84

Performing a two-sample hypothesis test for variance or standard deviation can be streamlined using the two-sample F test function available on graphing calculators like the TI-84. This test evaluates whether there is significant evidence to support a claim about the relationship between two population standard deviations, typically denoted as σ₁ and σ₂. The test involves comparing sample standard deviations and sample sizes to determine if one population variance is greater than the other.

When conducting this test, the null hypothesis (H₀) generally states that the two population standard deviations are equal, i.e., σ₁ = σ₂, while the alternative hypothesis (H₁) reflects the claim being tested, such as σ₁ > σ₂ for a right-tailed test. It is important to note that although the test concerns variances, hypotheses can be expressed in terms of standard deviations without squaring, depending on the context.

To perform the test using sample statistics, input the sample standard deviations (s₁ and s₂) and their corresponding sample sizes (n₁ and n₂) into the calculator’s two-sample F test function under the statistics tab. Ensure that s₁ corresponds to the larger sample standard deviation, as the F statistic is calculated as the ratio of the larger variance to the smaller variance. The F statistic is computed as:

\[F = \frac{s_1^2}{s_2^2}\]

The calculator then provides the F statistic and the p-value, which is used to make a decision. If the p-value is less than the significance level α (commonly 0.05), the null hypothesis is rejected, indicating sufficient evidence to support the claim that σ₁ is greater than σ₂. Conversely, if the p-value exceeds α, there is insufficient evidence to reject the null hypothesis.

When raw data is available instead of summary statistics, enter the data sets into separate lists (e.g., L1 and L2) on the calculator. Then, select the two-sample F test under the data tab, specify the lists, and choose the appropriate alternative hypothesis symbol. The calculator will compute the F statistic and p-value based on the data, allowing for the same decision-making process.

For example, if the calculated F statistic is 1.36 with a p-value of 0.026 and α = 0.05, the p-value being less than α leads to rejecting the null hypothesis, supporting the claim that σ₁ > σ₂. However, if the F statistic is 1.7 with a p-value of 0.286, the p-value exceeds α, so the null hypothesis is not rejected, indicating insufficient evidence to conclude that σ₁ is greater than σ₂.

Using the two-sample F test function simplifies the hypothesis testing process for variances by automating calculations and providing clear outputs for the F statistic and p-value. This approach enhances efficiency and accuracy when testing claims about population variances or standard deviations, whether working with summary statistics or raw data.

A historian is comparing the variation in weights of rare coins from different time periods. The data from two independent random samples is shown below. Using a 0.05 significance level and a graphing calculator, test the claim that the variation of weights before the 1900s is greater than after the 1900s.
$H_0$: _________ $H_a$: _________ $P$-value: ______
Because $P$-value [ < | > ] $\alpha$, we [ REJECT | FAIL TO REJECT ] $H_0$, there is [ ENOUGH | NOT ENOUGH ] evidence to suggest…

• A. Because $P$-value > $α$, we FAIL TO REJECT $H_0$, there is ENOUGH evidence to suggest $\sigma_1=\sigma_2$.
• B. Because $P$-value > $α$, we FAIL TO REJECT $H_0$, there is NOT ENOUGH evidence to suggest $\sigma_1=\sigma_2$.
• C. Because $P$-value < $α$, we REJECT $H_0$, there is ENOUGH evidence to suggest $\sigma_1>\sigma_2$.
• D. Because $P$-value < $α$, we REJECT $H_0$, there is NOT ENOUGH evidence to suggest $\sigma_1>\sigma_2$.

Answer: (C)

Hypothesis Test for Two Variances Using a TI-84 Example 1

When comparing the variation in delivery times between new and experienced drivers, a two-sample variance test can be used to determine if one group exhibits greater variability than the other. This test evaluates whether the standard deviation of delivery times for new drivers is significantly greater than that of experienced drivers. The process begins by formulating hypotheses: the null hypothesis assumes equal population standard deviations (\(\sigma_1 = \sigma_2\)), while the alternative hypothesis reflects the claim that the standard deviation for new drivers (\(\sigma_1\)) is greater than that for experienced drivers (\(\sigma_2\)), expressed as \(\sigma_1 > \sigma_2\).

Data from independent random samples of delivery times are entered into statistical software or a graphing calculator, such as the TI-84, using lists to organize the two groups. The two-sample F-test for variances is then selected, specifying the direction of the test to match the alternative hypothesis. The calculator computes the F-statistic, which is the ratio of the sample variances, and the corresponding p-value. For example, an F-statistic of 3.95 with a p-value of 0.041 indicates that the observed variation in delivery times for new drivers is statistically significantly greater than that for experienced drivers at the 0.05 significance level.

Since the p-value (0.041) is less than the chosen alpha level (0.05), the null hypothesis is rejected, providing sufficient evidence to support the claim that new drivers have greater variability in delivery times. This approach is essential in business contexts where understanding variability can impact operational decisions, such as driver training or scheduling. The two-sample variance test thus offers a robust method for comparing variability between independent groups, even when sample sizes differ.

Here’s what students ask on this topic:

How do you perform a two-sample F-test for variance using a graphing calculator?

To perform a two-sample F-test for variance on a graphing calculator, first determine if you have raw data or just sample statistics. If you have raw data, enter the data into two separate lists (e.g., L1 and L2). If you only have sample statistics, you will input the sample standard deviations and sample sizes directly. Next, access the Stat menu, navigate to the Tests tab, and select the two-sample F-test option. Choose the appropriate input method (data or stats). Enter the values for standard deviations (${}_{}$) and sample sizes (${}_{}$) accordingly. Specify the alternative hypothesis (e.g., ${}_{}>{}_{}$) and run the test. The calculator will output the F-statistic and p-value, which you use to decide whether to reject the null hypothesis based on your significance level (commonly 0.05).

What is the null and alternative hypothesis in a two-sample F-test for variance?

In a two-sample F-test for variance, the null hypothesis ($H\_0$) states that the population variances are equal: ${}_{}={}_{}$. The alternative hypothesis ($H\_a$) depends on the claim being tested. For example, if you want to test whether the variance of the first population is greater than the second, the alternative hypothesis is ${}_{}>{}_{}$. Sometimes, the test can be two-tailed (${}_{}\backslash neq{}_{}$) or left-tailed (), depending on the research question. Correctly stating these hypotheses is essential before performing the test.

How do you interpret the p-value from a two-sample F-test for variance on a graphing calculator?

The p-value from a two-sample F-test for variance indicates the probability of observing the test statistic or something more extreme if the null hypothesis is true. After running the test on your graphing calculator, compare the p-value to your chosen significance level ($\backslash alpha$, often 0.05). If the p-value is less than $\backslash alpha$, you reject the null hypothesis, concluding there is sufficient evidence that the variances differ as specified by the alternative hypothesis. If the p-value is greater than $\backslash alpha$, you fail to reject the null hypothesis, meaning there is not enough evidence to support the claim that the variances are different. This decision helps in making statistical inferences about population variances.

What steps should you follow when given raw data to perform a two-sample F-test for variance on a TI-84 calculator?

When given raw data, first enter the data sets into two lists (e.g., L1 and L2) by pressing Stat > Edit. Input the first sample data into L1 and the second into L2. Next, press Stat, navigate to the Tests menu, and select the two-sample F-test. Choose the Data tab and specify L1 and L2 as the data lists. Set the alternative hypothesis according to your test (e.g., ${}_{}>{}_{}$). Then, press Calculate to obtain the F-statistic and p-value. Use these results to make your conclusion about the variances.

Why must the larger sample variance be assigned as ${}_{}$ in a two-sample F-test?

In a two-sample F-test for variance, the test statistic is calculated as the ratio of the larger sample variance to the smaller one: $F\; =\; \backslash frac\{{}_{}\}\{{}_{}\}$. Assigning the larger variance as ${}_{}$ ensures that the F-statistic is always greater than or equal to 1, which simplifies interpretation and aligns with the F-distribution's properties. This convention helps avoid confusion and ensures the test statistic falls within the correct range for the F-distribution tables or calculator outputs, making hypothesis testing more straightforward.

How do you decide whether to reject or fail to reject the null hypothesis in a two-sample F-test for variance?

After performing the two-sample F-test for variance, compare the p-value to your significance level ($\backslash alpha$, commonly 0.05). If the p-value is less than $\backslash alpha$, reject the null hypothesis, indicating sufficient evidence that the population variances differ as stated in the alternative hypothesis. If the p-value is greater than or equal to $\backslash alpha$, fail to reject the null hypothesis, meaning there is not enough evidence to support a difference in variances. This decision guides whether the observed difference in sample variances is statistically significant or likely due to random chance.