Two Variances - Graphing Calculator: Videos & Practice Problems

Performing a two-sample F test for variance involves comparing sample 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. The null hypothesis typically states $\sigma 1_{}=\sigma 2_{}$, while the alternative reflects the claim, such as $\sigma 1_{}>\sigma 2_{}$. A p-value less than the significance level $\alpha$ leads to rejecting the null hypothesis, indicating evidence for the alternative. This method applies whether using raw data or summary statistics, enhancing efficiency in hypothesis testing for variance.

Hypothesis Test for Two Variances Using a TI-84

Performing a two-sample hypothesis test for variance or standard deviation can be efficiently done using the two-sample F test function available on graphing calculators like the TI-84. This test helps determine if there is significant evidence to support a claim about the relationship between two population standard deviations, typically denoted as σ₁ and σ₂. The test compares the variances of two independent samples by calculating the F statistic, which is the ratio of the larger sample variance to the smaller one, ensuring the test statistic is always greater than or equal to 1.

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. The significance level, α, commonly set at 0.05, determines the threshold for rejecting the null hypothesis based on the p-value obtained from the test.

For cases where only summary statistics are provided, such as sample standard deviations (s₁, s₂) and sample sizes (n₁, n₂), the two-sample F test can be performed by entering these values directly into the calculator’s statistics input mode. It is important to assign s₁ to the larger sample standard deviation to maintain the correct calculation of the F statistic as:

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

Here, s₁² and s₂² represent the sample variances. After inputting the values and selecting the appropriate test direction (greater than, less than, or not equal), the calculator outputs the F statistic and the p-value. If the p-value is less than α, the null hypothesis is rejected, indicating sufficient evidence to support the alternative hypothesis that σ₁ is greater than σ₂.

When raw data is available instead of summary statistics, the process involves entering the data sets into separate lists (e.g., L1 and L2) on the calculator. The two-sample F test is then performed by selecting the data input mode and specifying the lists containing the sample data. The hypotheses remain the same, and the calculator computes the F statistic and p-value based on the sample variances derived from the data. A p-value greater than α leads to failing to reject the null hypothesis, meaning there is insufficient evidence to conclude that σ₁ is greater than σ₂.

Understanding the interpretation of the p-value in the context of the significance level is crucial. The p-value represents the probability of observing an F statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis.

In summary, the two-sample F test is a powerful tool for comparing variances or standard deviations between two independent samples. Whether using summary statistics or raw data, the test involves calculating the F statistic as the ratio of sample variances, formulating hypotheses about the equality or inequality of population standard deviations, and making decisions based on the p-value relative to the chosen significance level. Mastery of this test enhances the ability to analyze variability in different populations effectively.

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 > $\alpha$, we FAIL TO REJECT $H_0$, there is ENOUGH evidence to suggest ${\sigma }_1={\sigma }_2$.
• B. Because $P$-value > $\alpha$, we FAIL TO REJECT $H_0$, there is NOT ENOUGH evidence to suggest ${\sigma }_1={\sigma }_2$.
• C. Because $P$-value < $\alpha$, we REJECT $H_0$, there is ENOUGH evidence to suggest ${\sigma }_1>{\sigma }_2$.
• D. Because $P$-value < $\alpha$, 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 resting heart rates (RHR) between nonathletes and athletes, a two-variance hypothesis test can determine if one group exhibits significantly higher variability than the other. This test is particularly useful when analyzing independent random samples with unequal sizes, as it assesses whether the population variances differ.

To begin, label the sample with the higher standard deviation as group one (s₁) and the other as group two (s₂). In this context, nonathletes typically have a higher standard deviation in RHR, making them s₁, while athletes are s₂. The hypotheses are then formulated based on the claim about the variances (σ₁² and σ₂²). The null hypothesis (H₀) assumes equal population variances:

\[H_0: \sigma_1^2 = \sigma_2^2\]

The alternative hypothesis (Hₐ) reflects the claim that the variance of nonathletes is greater than that of athletes:

\[H_a: \sigma_1^2 > \sigma_2^2\]

Using a significance level (α) of 0.05, the test statistic is calculated through an F-test, which compares the ratio of the sample variances. The F statistic is given by:

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

where \( s_1^2 \) and \( s_2^2 \) are the sample variances of groups one and two, respectively. This ratio follows an F-distribution with degrees of freedom corresponding to each sample size minus one.

To perform the test efficiently, data can be input into a graphing calculator, such as the TI-84, by entering the raw data into separate lists for each group. The calculator’s statistical test functions allow selection of the sample F-test, specifying the data lists and the direction of the alternative hypothesis (greater than). The output provides the F statistic and the p-value.

For example, an F statistic of 3.95 with a p-value of 0.041 indicates that the p-value is less than the significance level (0.05). This leads to rejecting the null hypothesis, providing sufficient evidence to support the claim that the variance in resting heart rate among nonathletes is greater than that of athletes. This conclusion aligns with the understanding that athletes typically have more consistent and conditioned heart rates due to their physical training.

Overall, the two-variance hypothesis test is a powerful method for comparing variability between two independent groups, especially when sample sizes differ. It relies on understanding population variances, formulating appropriate hypotheses, calculating the F statistic, and interpreting the p-value to make informed conclusions about differences in variation.

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. Choose the appropriate input method (data or stats). Enter the values for $s_1$, $s_2$, $n_1$, and $n_2$. Specify the alternative hypothesis (e.g., ${\sigma }_1>{\sigma }_2$). Then, calculate to obtain the F statistic and p-value. Compare the p-value to your significance level $\alpha$ to decide whether to reject the null hypothesis that the variances are equal.

What are the hypotheses in a two-sample F test for comparing variances?

In a two-sample F test for variances, the null hypothesis ($H{0}_{}$) states that the population variances are equal: ${\sigma }_1={\sigma }_2$. The alternative hypothesis ($H{a}_{}$) reflects the claim you want to test, such as ${\sigma }_1>{\sigma }_2$ if you suspect the first variance is greater, or ${\sigma }_1<{\sigma }_2$ if less. Sometimes, a two-tailed test is used with ${\sigma }_1\ne {\sigma }_2$. These hypotheses guide the test and determine the rejection region based on the F distribution.

How do you interpret the p-value in a two-sample F test for variance?

The p-value in a two-sample F test for variance indicates the probability of observing an F statistic as extreme as, or more extreme than, the one calculated from your samples, assuming the null hypothesis is true. If the p-value is less than the chosen significance level $\alpha$ (commonly 0.05), you reject the null hypothesis, concluding there is significant evidence that the variances differ as stated in the alternative hypothesis. If the p-value is greater than $\alpha$, you fail to reject the null hypothesis, meaning there is insufficient evidence to support a difference in variances. This helps in making decisions about population variability based on sample data.

What is the importance of ensuring $s_{1}2$ is the larger sample variance in the two-sample F test?

In the two-sample F test, the test statistic is calculated as the ratio of the larger sample variance to the smaller sample variance: $F=\frac{s_1}{2}{s}_{2}2$. Ensuring $s_{1}2$ is the larger variance standardizes the test statistic to be greater than or equal to 1, which aligns with the F distribution's properties. This practice simplifies interpretation and comparison with critical values or p-values. If the variances are not ordered correctly, the test statistic might be less than 1, complicating the analysis and potentially leading to incorrect conclusions.

Can you perform a two-sample F test for variance using only sample statistics without raw data?

Yes, you can perform a two-sample F test for variance using only sample statistics such as sample standard deviations and sample sizes. Many graphing calculators have a function for the two-sample F test that allows you to input these statistics directly. You enter the sample standard deviations $s_1$ and $s_2$, along with the corresponding sample sizes $n_1$ and $n_2$. The calculator then computes the F statistic and p-value, enabling hypothesis testing without needing the raw data. This method is efficient and useful when only summary statistics are available.

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

To enter raw data for a two-sample F test on a TI-84 calculator, follow these steps: 1) Press the Stat button and select Edit to open the list editor. 2) Enter the first sample's data into list L1 by typing each value and pressing Enter. 3) Enter the second sample's data into list L2 similarly. 4) After entering data, press Stat again, navigate to the Tests menu, and select 2-SampFTest. 5) Choose the Data input option, specify L1 and L2 as the data lists, and select the alternative hypothesis. 6) Press Calculate to obtain the F statistic and p-value. This process allows you to test variance differences using raw data efficiently.