Comparing Means: Hypothesis Test With Sample Data

Hey there, math enthusiasts! Today, we're diving into a statistical adventure, specifically a hypothesis test to figure out if the average of one population is less than the average of another. We'll be using sample data and assuming the populations follow a normal distribution. Buckle up, it's going to be fun! This exploration revolves around testing the hypothesis that the population mean of one group is significantly smaller than the population mean of another. We're going to use sample data to make an inference about the broader populations. This is super useful in real-world scenarios – think comparing the effectiveness of two different medications, or the performance of two different teaching methods, or even the sales generated by two different marketing strategies. Understanding how to compare means allows us to make evidence-based decisions. The core idea is to use the sample data to calculate a test statistic. This test statistic measures how far apart the sample means are from each other, in relation to the variability within the samples. Based on this test statistic, we then determine a p-value. The p-value is the probability of observing our sample results, or even more extreme results, if the null hypothesis is true. Finally, we compare the p-value to our significance level (alpha). If the p-value is less than alpha, we reject the null hypothesis and conclude that there is a statistically significant difference between the population means. Specifically, we're trying to figure out if the mean of population 1 ($\mu_1$) is less than the mean of population 2 ($\mu_2$). We will use the provided sample data along with a set significance level. We are going to assume that the populations are normally distributed. Let's break down the whole process step-by-step. Let's get started, shall we?

Setting the Stage: The Hypothesis and Significance Level

Alright, before we jump into the calculations, let's set the stage. Our main goal is to test the hypothesis that the population mean of the first sample ($\mu_1$) is less than the population mean of the second sample ($\mu_2$). In statistical terms, we have to set up our null and alternative hypotheses. The null hypothesis (H0) is always the status quo, the assumption we're trying to disprove. In this case, our null hypothesis is that there is no difference or the mean of the first population is greater than or equal to the mean of the second population ($\mu_1 \ge \mu_2$). The alternative hypothesis (Ha), which is what we're trying to prove, states that the mean of the first population is, in fact, less than the mean of the second population ($\mu_1 < \mu_2$). This is a one-tailed test because we are only interested in whether $\mu_1$ is less than $\mu_2$. The other side, whether $\mu_1$ is greater than $\mu_2$ is not our focus. Now, we also have to set our significance level, often denoted as alpha ($\alpha$). This value represents the probability of rejecting the null hypothesis when it's actually true (a Type I error). We are given an alpha = 0.01. This means we are willing to accept a 1% chance of making a mistake. This significance level gives us a threshold for our p-value. If the p-value (which we'll calculate later) is less than 0.01, then we reject the null hypothesis. It is very important to clearly define what we are trying to test and the conditions under which we will decide the test. Having a good understanding of these initial steps is essential to performing an accurate and reliable hypothesis test.

Now we've got our hypotheses and the significance level. Let's move on to the actual calculations!

Crunching the Numbers: Calculations and Formulas

Okay, time for some number-crunching! We'll need some additional information to proceed with our hypothesis test. We will need the sample means and sample standard deviations. Unfortunately, those weren't provided, so we're going to use an example. Let's assume the following sample data (because we didn't get them from you, sorry!): Let's say, we have:

• Sample 1: n1 = 31, mean = 20, standard deviation, s1 = 2
• Sample 2: n2 = 25, mean = 22, standard deviation, s2 = 3

We're dealing with a two-sample t-test because we're comparing the means of two independent samples and don't know the population standard deviations. Here's the formula for the t-statistic:

• t = (x̄1 - x̄2) / √((s1²/n1) + (s2²/n2))

Where:

• x̄1 is the sample mean of sample 1
• x̄2 is the sample mean of sample 2
• s1 is the sample standard deviation of sample 1
• s2 is the sample standard deviation of sample 2
• n1 is the sample size of sample 1
• n2 is the sample size of sample 2

Let's plug in our example numbers:

t = (20 - 22) / √((2²/31) + (3²/25)) = -2 / √(0.129 + 0.36) = -2 / √0.489 = -2 / 0.699 = -2.86

So, our t-statistic is -2.86. Next, we need to calculate the degrees of freedom (df). Because we have two independent samples, we typically use the following formula. But note that depending on your data, there might be other ways of doing this calculation.

• df = n1 + n2 - 2 = 31 + 25 - 2 = 54

With these calculations in hand, we will be able to get a p-value.

Deciding What It All Means: P-value and Conclusion

Great! We have our test statistic (t = -2.86) and degrees of freedom (df = 54). Now it is time to find the p-value. The p-value tells us the probability of observing a test statistic as extreme as, or more extreme than, the one we calculated, assuming the null hypothesis is true. To find the p-value, we can use a t-table or a statistical software package (like R, Python, or even an online calculator). Given our t-statistic of -2.86 and 54 degrees of freedom, the p-value is approximately 0.003. Since this is a one-tailed test (because our alternative hypothesis is μ1 < μ2), we only need to consider the tail in the direction of our alternative hypothesis. The p-value is very small. Remember our significance level, alpha = 0.01? Well, our p-value (0.003) is less than alpha (0.01). That is a significant result! Because the p-value (0.003) is less than our significance level (0.01), we reject the null hypothesis. We have enough evidence to conclude that the mean of the first population is indeed less than the mean of the second population. In practical terms, this means that, based on our sample data, there is a statistically significant difference between the two populations, and the first population mean is likely smaller. This kind of analysis is very important. Whether we are trying to determine if one drug is better than another, or which business strategy to pursue, this information is very useful.

Important Considerations

Now, a couple of important things to keep in mind! First, the results of our test are based on the sample data we used. Different samples might lead to slightly different results. Second, we assumed that our populations were normally distributed. If this assumption is badly violated, then our results might not be reliable. Third, statistical significance doesn't necessarily mean practical significance. A small difference between the means might be statistically significant with a large sample size, but it might not be meaningful in the real world. Finally, always consider the context of your data! Statistical analysis is a tool, but it should always be combined with your knowledge of the subject matter.

Conclusion

We successfully tested the hypothesis that μ1 < μ2 using a significance level of 0.01. We started by setting up our hypotheses, then we used the sample data (after getting some!) to calculate the test statistic and the p-value. We then compared the p-value to our alpha level. Because the p-value was smaller than alpha, we rejected the null hypothesis and concluded that the mean of the first population is, in fact, less than the mean of the second population. Keep practicing, keep learning, and you'll become a stats whiz in no time!