10. Hypothesis Testing for Two Samples

Two Proportions Hypothesis Test - Excel

10. Hypothesis Testing for Two Samples

Two Proportions Hypothesis Test - Excel: Videos & Practice Problems

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Performing a hypothesis test for two population proportions in Excel involves defining the null hypothesis (equal proportions) and alternative hypothesis (e.g., $p 1 < p 2$ ), calculating sample proportions $p_{^1}$ and $p_{^2}$ , and computing the pooled proportion $p ¯$ . The test statistic z-score is calculated using $z = \frac{p_{^1} - p_{^2}}{\sqrt{\frac{p}{¯} + \frac{p}{¯}}}$ . Excel functions like COUNTIF and NORM.S.DIST assist in calculating counts and p-values. Comparing the p-value to alpha (significance level) determines whether to reject the null hypothesis, guiding conclusions about population proportions. This method integrates inferential statistics and hypothesis testing principles effectively.

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Two Proportions Hypothesis Test - Excel

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Two Proportions Hypothesis Test - Excel Video Summary

Performing a hypothesis test for two population proportions in Excel involves following the standard four-step process, even though there isn't a single built-in function to automate the entire test. The goal is to determine if there is a significant difference between two proportions, such as the proportion of customers ordering muffins on weekdays versus weekends.

First, establish the null hypothesis ($H_0$) that the two population proportions are equal, $p_1 = p_2$, where $p_1$ represents the weekday proportion and $p_2$ the weekend proportion. The alternative hypothesis ($H_a$) reflects the claim being tested; for example, $p_1 < p_2$ if the belief is that the weekday proportion is lower.

Next, calculate the test statistic using the z-score formula for two proportions:

\[z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\bar{p} \bar{q} \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}\]

Here, $\hat{p}_1$ and $\hat{p}_2$ are the sample proportions, $n_1$ and $n_2$ are the sample sizes, and $\bar{p}$ is the pooled proportion calculated as:

\[\bar{p} = \frac{x_1 + x_2}{n_1 + n_2}\]

with $\bar{q} = 1 - \bar{p}$. The values $x_1$ and $x_2$ represent the counts of successes (e.g., orders with muffins) in each sample.

In Excel, use the COUNTIF function to find $x_1$ and $x_2$ by counting occurrences of the desired outcome (e.g., "yes" for muffin orders) within each data range. Then, determine the sample proportions by dividing these counts by their respective sample sizes.

Calculate the pooled proportion $\bar{p}$ and its complement $\bar{q}$, then compute the numerator ($\hat{p}_1 - \hat{p}_2$) and the denominator (the square root term) separately to reduce complexity and minimize errors. Finally, compute the z-score by dividing the numerator by the denominator.

To find the p-value corresponding to the z-score, use Excel’s NORM.S.DIST function with the cumulative option set to TRUE for left-tail probabilities:

\[\text{p-value} = \text{NORM.S.DIST}(z, \text{TRUE})\]

For right-tail or two-tailed tests, adjust accordingly using complementary probabilities or doubling the tail area.

Compare the p-value to the significance level $\alpha$ (commonly 0.05). If the p-value is greater than $\alpha$, fail to reject the null hypothesis, indicating insufficient evidence to support the alternative claim. Conversely, if the p-value is less than or equal to $\alpha$, reject the null hypothesis, suggesting a statistically significant difference between the two proportions.

This methodical approach in Excel not only streamlines calculations but also reinforces understanding of hypothesis testing concepts for two population proportions, making it a valuable skill for analyzing categorical data in real-world contexts.

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Two Proportions Hypothesis Test - Excel Example 1

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Two Proportions Hypothesis Test - Excel Example 1 Video Summary

When comparing proportions between two independent groups, such as middle school and high school students' exam scores, hypothesis testing helps determine if there is a significant difference. In this scenario, the goal is to test whether the proportion of middle school students who exceed expectations (denoted as p₁) is greater than the proportion of high school students who do so (p₂), using a significance level of α = 0.1.

The null hypothesis (H₀) assumes no difference between the proportions: $H_0: p_1 = p_2$. The alternative hypothesis (Hₐ) reflects the claim that the middle school proportion is higher: $H_a: p_1 > p_2$.

To perform the test, sample data from 50 students in each group is analyzed. The number of students scoring "exceeding expectations" (denoted as x) is counted for each group, yielding $x_1 = 17$ for middle school and $x_2 = 15$ for high school. The sample proportions ($\hat{p}_1$ and $\hat{p}_2$) are calculated by dividing these counts by the sample sizes:

\[\hat{p}_1 = \frac{x_1}{n_1} = \frac{17}{50} = 0.34\]\[\hat{p}_2 = \frac{x_2}{n_2} = \frac{15}{50} = 0.30\]

Next, the pooled sample proportion ($\bar{p}$) combines successes from both samples to estimate the common proportion under the null hypothesis:

\[\bar{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{17 + 15}{50 + 50} = 0.32\]

The complement of the pooled proportion is $\bar{q} = 1 - \bar{p} = 0.68$.

The test statistic is a z-score calculated by comparing the difference between sample proportions to the standard error of that difference:

\[z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\bar{p} \bar{q} \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}\]

Breaking down the denominator, the variance components for each sample are:

\[\frac{\bar{p} \bar{q}}{n_1} = \frac{0.32 \times 0.68}{50} = 0.00435\]\[\frac{\bar{p} \bar{q}}{n_2} = \frac{0.32 \times 0.68}{50} = 0.00435\]

Summing these and taking the square root gives the standard error:

\[\sqrt{0.00435 + 0.00435} = \sqrt{0.0087} \approx 0.0933\]

Thus, the z-score is:

\[z = \frac{0.34 - 0.30}{0.0933} \approx 0.43\]

To find the p-value corresponding to this z-score for a right-tailed test, the cumulative distribution function (CDF) of the standard normal distribution is used. Since the CDF gives the left-tail probability, the right-tail p-value is calculated as:

\[p\text{-value} = 1 - \Phi(z)\]

For $z = 0.43$, the p-value is approximately 0.33.

Comparing the p-value to the significance level α = 0.1, since 0.33 > 0.1, there is insufficient evidence to reject the null hypothesis. This means the data do not support the claim that the proportion of middle school students exceeding expectations is higher than that of high school students.

Understanding this process highlights the importance of formulating clear hypotheses, calculating sample proportions, pooling data appropriately, and interpreting z-scores and p-values within the context of hypothesis testing for proportions.

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Two Proportions Hypothesis Test - Excel Example 2

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Two Proportions Hypothesis Test - Excel Example 2 Video Summary

When comparing the proportions of senior varsity athletes between two high schools, a two-proportion hypothesis test can determine if the proportions are statistically different. The null hypothesis assumes that the proportions are equal, expressed as $H_0: p_1 = p_2$, where $p_1$ and $p_2$ represent the proportions of senior varsity athletes at School A and School B, respectively. The alternative hypothesis, $H_a: p_1 \neq p_2$, tests whether these proportions differ without specifying a direction.

To conduct this test, data is collected from random samples of 50 varsity athletes from each school, recording the number of seniors in each group. The sample proportions $\hat{p}_1$ and $\hat{p}_2$ are calculated by dividing the number of seniors by the total sample size for each school. For example, if School A has 19 seniors out of 50 athletes, then $\hat{p}_1 = \frac{19}{50} = 0.38$, and if School B has 32 seniors out of 50, then $\hat{p}_2 = \frac{32}{50} = 0.64$.

Next, the pooled proportion $\bar{p}$ is computed to combine the successes from both samples, using the formula:

\[\bar{p} = \frac{x_1 + x_2}{n_1 + n_2}\]

where $x_1$ and $x_2$ are the counts of seniors in each sample, and $n_1$ and $n_2$ are the respective sample sizes. The complement $q$ is calculated as $q = 1 - \bar{p}$.

The test statistic $z$ is then calculated using the formula for two-proportion z-tests:

\[z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\bar{p} \cdot q \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}\]

This formula measures how many standard errors the observed difference in sample proportions is from the hypothesized difference of zero under the null hypothesis.

After calculating the $z$-score, the corresponding two-tailed p-value is found by determining the probability of observing a value as extreme or more extreme than the test statistic in either tail of the standard normal distribution. This is done by finding the cumulative probability for the smaller tail (left tail if $z$ is negative, right tail if positive) and doubling it:

\[p\text{-value} = 2 \times P(Z \leq z) \quad \text{if } z < 0\]

For example, a $z$-score of approximately -2.6 yields a p-value around 0.009. Comparing this p-value to the significance level $\alpha = 0.01$, since \$0.009 < 0.01$, the null hypothesis is rejected. This indicates sufficient evidence to conclude that the proportions of senior varsity athletes differ between the two schools.

Understanding this process is essential for analyzing categorical data across groups, applying hypothesis testing principles, and interpreting statistical significance in real-world contexts.

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10. Hypothesis Testing for Two Samples

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To perform a hypothesis test for two population proportions in Excel, start by defining your null hypothesis (usually that the two proportions are equal) and your alternative hypothesis (such as one proportion being less than the other). Next, calculate the sample proportions by dividing the number of successes by the sample size for each group. Then, compute the pooled proportion $p = x$ ₁ + x₂n₁ + n₂. Use this to find the standard error and calculate the z-score with $z =$ ₁ − ₂p×q (1n₁ + 1n₂). Finally, use Excel's NORM.S.DIST function to find the p-value from the z-score and compare it to your significance level to decide whether to reject the null hypothesis.

When conducting a two proportions hypothesis test in Excel, several functions are particularly helpful. The COUNTIF function counts the number of successes (e.g., orders with muffins) in each sample by searching for a specific criterion like "yes." For example, =COUNTIF(range, "yes") counts how many times "yes" appears in the data range. To calculate the p-value from the z-score, use =NORM.S.DIST(z, TRUE), which returns the cumulative probability for the standard normal distribution. These functions simplify the calculation process, allowing you to efficiently compute sample counts and p-values without manual calculations.

The pooled proportion, denoted as $p$ , combines the successes from both samples and is essential for calculating the standard error in a two proportions hypothesis test. In Excel, calculate it by adding the number of successes from both samples and dividing by the total sample size. The formula is $p = x$ ₁ + x₂n₁ + n₂. For example, if x1 and x2 are in cells A1 and A2, and n1 and n2 are in B1 and B2, you can enter = (A1 + A2) / (B1 + B2) to get the pooled proportion.

In a two proportions hypothesis test, the p-value represents the probability of observing the test statistic or something more extreme if the null hypothesis is true. After calculating the p-value in Excel using the NORM.S.DIST function, compare it to your chosen significance level (alpha, often 0.05). If the p-value is less than alpha, you reject the null hypothesis, indicating sufficient evidence to support the alternative hypothesis. If the p-value is greater than alpha, you fail to reject the null hypothesis, meaning there is not enough evidence to conclude a difference between the two population proportions. This decision guides your conclusion about the populations.

To calculate the z-score for two population proportions in Excel, follow these steps: First, find the number of successes ( $x$ ₁ and $x$ ₂) using COUNTIF. Then, determine the sample sizes ( $n$ ₁ and $n$ ₂). Calculate the sample proportions ₁ = x_1 / n_1 and ₂ = x_2 / n_2. Next, compute the pooled proportion $p$ and its complement $q = 1 - p$ . Calculate the standard error as $\sqrt{p × q × (1/n_1 + 1/n_2)}$ . Finally, compute the z-score with $z = ($ ₁ - ₂) / ext{standard error}. This z-score can then be used to find the p-value.

Setting up hypotheses for a two proportions test in Excel begins with defining the null hypothesis ( $H_0$ ) and the alternative hypothesis ( $H_a$ ). The null hypothesis usually states that the two population proportions are equal: $H_0: p_1 = p_2$ . The alternative hypothesis depends on the research question and can be one-sided or two-sided. For example, if testing whether the proportion in group 1 is less than group 2, use $H_a: p_1 < p_2$ . These hypotheses guide the direction of the test and the calculation of the p-value. You then proceed with calculations in Excel to test these hypotheses.