Hypothesis Testing: Proportions - Excel: Videos & Practice Problems

Performing a hypothesis test for a population proportion involves stating the null hypothesis $H\_0:p=0.25$ and alternative hypothesis $H\_a:p<0.25$. Calculate the sample proportion $p^$ using the count of successes divided by sample size. The test statistic z-score is computed by $z=\frac{p^-p}{\sqrt{p\cdot (1-p)/n}}$. Use the standard normal cumulative distribution function to find the p-value. Comparing p-value to alpha level determines if the null hypothesis is rejected, guiding conclusions about population proportions in statistical inference.

Performing Hypothesis Tests: Proportions - Excel

Performing a hypothesis test for a population proportion in Excel involves a systematic approach that includes writing hypotheses, calculating the test statistic, determining the p-value, and drawing a conclusion. Suppose a town surveys 30 citizens to estimate the proportion of voters needing an absentee ballot, with a previous election proportion of 25%. The town suspects the current proportion might be less than 25%, prompting a hypothesis test at a significance level of α = 0.05.

First, the null hypothesis (H₀) states that the population proportion p equals 0.25, while the alternative hypothesis (H₁) claims that p is less than 0.25. To analyze the survey data, Excel’s COUNTIF function efficiently counts the number of "yes" responses, representing citizens needing absentee ballots. For example, if there are 4 "yes" responses out of 30, the sample size n is 30, and the sample proportion p̂ is calculated as \(p̂ = \frac{4}{30} \approx 0.133\).

The test statistic, or z-score, is computed using the formula:

\[ z = \frac{p̂ - p}{\sqrt{\frac{p(1-p)}{n}}} \]

where p is the claimed population proportion (0.25), p̂ is the sample proportion, and n is the sample size. In Excel, this formula can be entered carefully to ensure accuracy, using cell references for each value and the SQRT function for the square root.

Once the z-score is obtained (e.g., approximately -1.48), the next step is to find the p-value, which represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. Since the alternative hypothesis is one-sided (less than), the p-value corresponds to the left-tail probability of the standard normal distribution. Excel’s NORM.S.DIST function calculates this cumulative probability:

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

For a z-score of -1.48, the p-value is approximately 0.07.

Comparing the p-value to the significance level α = 0.05 determines the conclusion. Because 0.07 > 0.05, there is insufficient evidence to reject the null hypothesis. This means the data do not support the claim that the proportion of voters needing absentee ballots is less than 25%. Understanding this process enhances skills in statistical inference, hypothesis testing, and the practical use of Excel functions for data analysis.

Performing Hypothesis Tests: Proportions - Excel Example 1

When testing a claim about a population proportion, such as whether more than 70% of citizens in a city own at least one pet, the process begins by formulating the hypotheses. The null hypothesis (denoted as H₀) states that the true population proportion p equals 0.7, reflecting the previous study's claim. The alternative hypothesis (H_a) represents the city official's suspicion that the true proportion is higher, so it is set as p > 0.7. This setup defines a right-tailed hypothesis test.

Next, the sample data is analyzed. With a sample size n = 200, the sample proportion \hat{p} is calculated by dividing the number of citizens who own pets by the total sample size. For example, if 156 out of 200 citizens own pets, the sample proportion is \hat{p} = 156/200 = 0.78. This value is crucial for computing the test statistic.

The test statistic for a proportion hypothesis test is a z-score, which measures how many standard deviations the sample proportion is from the hypothesized population proportion. The formula for the z-score is:

\[z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}\]

Here, p is the hypothesized population proportion (0.7), \hat{p} is the sample proportion (0.78), and n is the sample size (200). The denominator represents the standard error of the proportion, calculated as the square root of p(1-p)/n. Substituting the values, the z-score quantifies the deviation of the observed sample proportion from the null hypothesis.

Once the z-score is computed (approximately 2.47 in this case), it is converted into a p-value, which indicates the probability of observing a sample proportion as extreme as or more extreme than the one obtained, assuming the null hypothesis is true. Since the alternative hypothesis is one-sided (greater than), the p-value corresponds to the right-tail probability of the z-distribution. Because standard functions often provide the left-tail cumulative probability, the right-tail p-value is found by subtracting the left-tail probability from 1:

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

where \Phi(z) is the cumulative distribution function (CDF) of the standard normal distribution evaluated at the z-score.

Comparing the p-value to the significance level \alpha = 0.01 determines the test outcome. If the p-value is less than alpha, the null hypothesis is rejected, providing sufficient evidence to support the alternative hypothesis. In this example, the p-value (~0.007) is less than 0.01, so the conclusion is to reject the null hypothesis and infer that the true proportion of pet owners in the city is likely greater than 70%.

This hypothesis testing framework is essential for making data-driven decisions about population parameters, using sample data to assess claims with a controlled risk of error defined by the significance level.

Performing Hypothesis Tests: Proportions - Excel Example 2

An online retailer wants to determine if a new returns policy has affected the percentage of orders involving customer returns. Before the policy change, approximately 9.8% of orders included at least one returned product. To test this, a hypothesis test is conducted with a significance level of α = 0.1, using a random sample of 150 orders.

The first step is to establish the hypotheses. The null hypothesis (H₀) states that the true population proportion p remains equal to 0.098, reflecting no change in the return rate. The alternative hypothesis (Hₐ) posits that p is not equal to 0.098, indicating a possible change in the return percentage due to the new policy. This sets up a two-tailed test since the direction of change is not specified.

Next, the sample proportion is calculated by counting the number of orders with returns ("yes" responses) and dividing by the total sample size of 150. This proportion is found to be 0.08. With the sample proportion, expected population proportion, and sample size identified, the test statistic z is computed using the formula:

\[ z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} \]

where p is the expected population proportion (0.098), n is the sample size (150), and 𝑝̂ is the sample proportion (0.08). Substituting these values yields a z-score of approximately -0.74.

To determine the p-value for this two-tailed test, the cumulative probability corresponding to the z-score is found using the standard normal distribution. Since the z-score is negative, the left-tail probability is smaller. The two-tailed p-value is twice this left-tail probability, calculated as:

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

Using statistical software or functions, the p-value is approximately 0.4584.

Comparing the p-value to the significance level α = 0.1, since 0.4584 > 0.1, there is insufficient evidence to reject the null hypothesis. This means the data do not support a significant change in the proportion of orders with returns after the new policy was enacted. Therefore, it is concluded that the true population proportion of returned orders remains statistically unchanged at 9.8%.

Here’s what students ask on this topic:

How do you perform a hypothesis test for a population proportion in Excel?

To perform a hypothesis test for a population proportion in Excel, start by stating your null hypothesis (H₀) and alternative hypothesis (H_a). For example, H₀: $p=0.25$ and H_a: $p<0.25$. Next, calculate the sample proportion ($\mathrm{p̂}$) by dividing the number of successes by the sample size. Use the COUNTIF function to count successes, e.g., =COUNTIF(range, "yes"). Then compute the z-score using the formula $z=\frac{\mathrm{p̂}-p}{\sqrt{p(1-p)/n}}$. Finally, find the p-value with =NORM.S.DIST(z, TRUE) for a left-tailed test. Compare the p-value to your alpha level to decide whether to reject H₀.

What is the formula for calculating the z-score in a hypothesis test for proportions?

The z-score formula for a hypothesis test of a population proportion is given by $z=\frac{\mathrm{p̂}-p}{\sqrt{p(1-p)/n}}$, where $\mathrm{p̂}$ is the sample proportion, $p$ is the claimed population proportion under the null hypothesis, and $n$ is the sample size. This formula measures how many standard errors the sample proportion is away from the claimed proportion, helping determine if the observed data is consistent with the null hypothesis.

How do you find the p-value from a z-score in Excel for a hypothesis test on proportions?

In Excel, to find the p-value from a z-score for a hypothesis test on proportions, use the function =NORM.S.DIST(z, TRUE). This function returns the cumulative probability from the left tail up to the z-score. For a left-tailed test (alternative hypothesis $p<p$₀), this p-value directly represents the probability of observing a test statistic as extreme or more extreme than the calculated z-score. For right-tailed or two-tailed tests, you may need to adjust the p-value accordingly.

What are the steps to write a conclusion after performing a hypothesis test for proportions in Excel?

After calculating the p-value in Excel, compare it to your significance level (alpha). If the p-value is less than alpha, reject the null hypothesis (H₀), indicating sufficient evidence to support the alternative hypothesis (H_a). If the p-value is greater than alpha, fail to reject H₀, meaning there is not enough evidence to support H_a. For example, if $p=0.07$ and $\alpha =0.05$, since $p>\alpha$, you fail to reject H₀. Your conclusion should clearly state this decision and interpret it in the context of the problem.

How do you use the COUNTIF function in Excel to analyze survey data for hypothesis testing?

The COUNTIF function in Excel helps count the number of times a specific response appears in survey data, which is essential for calculating the sample proportion in hypothesis testing. The syntax is =COUNTIF(range, "criteria"). For example, if you want to count how many respondents answered "yes" to needing an absentee ballot, you select the data range and set the criteria as "yes" with quotes. This count gives the number of successes, which you then divide by the total sample size to find the sample proportion ($\mathrm{p̂}$).

What does it mean to fail to reject the null hypothesis in a hypothesis test for proportions?

Failing to reject the null hypothesis means that the sample data does not provide strong enough evidence to support the alternative hypothesis. In the context of a proportion test, it implies that the observed sample proportion is not significantly different from the claimed population proportion at the chosen significance level. For example, if the p-value is greater than alpha, you conclude there is insufficient evidence to say the population proportion differs from the hypothesized value. This does not prove the null hypothesis true, only that the data does not contradict it.