Hypothesis Test For Population Mean: A Step-by-Step Guide

Nov 10, 2025 by Andrew McMorgan 58 views

Hypothesis Testing for Population Mean: A Step-by-Step Guide

Hey Plastik Magazine readers! Ever found yourself staring at a statistical problem and feeling totally lost? Don't worry, we've all been there. Let's break down a common scenario in statistics: hypothesis testing for a population mean when you don't know the population standard deviation. Sounds intimidating, right? It's not as bad as it seems! We're going to walk through it together, step by step, using a real example. So, grab your thinking caps, and let's dive in!

Understanding the Problem

In this article, we're tackling a specific type of hypothesis test. Imagine you want to test a claim ( $H_a$ ) at a significance level of $\alpha = 0.002$ . This significance level is super important because it tells us how much risk we're willing to take of being wrong when we reject the null hypothesis.

Our hypotheses are as follows:

Null Hypothesis ( $H_0$ ): $\mu = 90.8$ (This is the statement we're trying to disprove. It's like the status quo.)
Alternative Hypothesis ( $H_a$ ): $\mu < 90.8$ (This is the claim we're trying to support with our data. It suggests the population mean is less than 90.8.)

We're also told that the population is normally distributed, which is fantastic news because it allows us to use certain statistical tests. However, there's a catch: we don't know the population standard deviation. This is where the t-test comes in handy, which we'll discuss later.

To make things concrete, we have some sample data:

83
91.9
81
78.3
83.4

This sample represents a small snapshot of the larger population we're interested in. Now, let's get into the nitty-gritty of how to tackle this problem. First, we'll start with part (a), which likely involves figuring out the appropriate test statistic.

Step 1: Identifying the Appropriate Test Statistic

Okay, guys, so the first big question is: which test do we use? When we're testing a hypothesis about a population mean ( $\mu$ ) and we don't know the population standard deviation, the t-test is our go-to friend. This is super important to remember! The t-test is designed precisely for situations like this, where we have to estimate the population standard deviation from the sample.

Why can't we just use a z-test, which is used when we do know the population standard deviation? Well, the t-test accounts for the extra uncertainty that comes from estimating the standard deviation. It has a slightly different distribution (the t-distribution) that has heavier tails than the standard normal distribution (which the z-test uses). This means the t-test is more conservative, which is a good thing when we're not entirely sure about the population standard deviation.

So, the test statistic we'll be using is the t-statistic. Now, how do we calculate it? The formula for the t-statistic is:

$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

Let's break this down:

$\bar{x}$ is the sample mean (the average of our data).
$\mu_0$ is the hypothesized population mean (from our null hypothesis, $H_0$ ).
$s$ is the sample standard deviation (a measure of the spread of our data).
$n$ is the sample size (the number of data points in our sample).

To calculate the t-statistic, we first need to find the sample mean ( $\bar{x}$ ) and the sample standard deviation ( $s$ ) from our data. Let's do that next!

Step 2: Calculating the Sample Mean and Standard Deviation

Alright, time to get our calculators (or spreadsheets!) ready. We need to crunch some numbers to find the sample mean and sample standard deviation. These are crucial for calculating our t-statistic.

First, let's tackle the sample mean ( $\bar{x}$ ). This is simply the average of our data points. Remember our data?

83
91.9
81
78.3
83.4

To find the mean, we add them up and divide by the number of data points (which is 5):

$\bar{x} = \frac{83 + 91.9 + 81 + 78.3 + 83.4}{5} = \frac{417.6}{5} = 83.52$

So, our sample mean is 83.52. Not too shabby!

Now, for the sample standard deviation ( $s$ ). This one's a bit more involved, but don't worry, we'll take it step by step. The formula for the sample standard deviation is:

$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}$

Okay, let's break this down piece by piece:

For each data point ( $x_i$ ), we subtract the sample mean ( $\bar{x}$ ). This gives us the deviation of each point from the mean.
We square each of these deviations. This gets rid of negative signs and emphasizes larger deviations.
We add up all the squared deviations. This is what the $\sum$ symbol means.
We divide this sum by $n-1$ , where $n$ is the sample size. This is called the degrees of freedom, and we use $n-1$ instead of $n$ to get an unbiased estimate of the population standard deviation.
Finally, we take the square root of the result. This brings us back to the original units of measurement.

Let's do the calculations. It's helpful to organize this in a table:

Data Point ( $x_i$ )	$x_i - \bar{x}$	$(x_i - \bar{x})^2$
83	-0.52	0.2704
91.9	8.38	70.2244
81	-2.52	6.3504
78.3	-5.22	27.2484
83.4	-0.12	0.0144
Total		104.108

So, the sum of the squared deviations is 104.108. Now we can plug this into the formula:

$s = \sqrt{\frac{104.108}{5-1}} = \sqrt{\frac{104.108}{4}} = \sqrt{26.027} \approx 5.10$

Therefore, our sample standard deviation is approximately 5.10. Whew! That was a bit of work, but we got there. Now we have both the sample mean and the sample standard deviation, so we're ready to calculate the t-statistic. Let's move on to the next step!

Step 3: Calculate the T-Statistic

Alright, we've got the sample mean ( $\bar{x} = 83.52$ ) and the sample standard deviation ( $s \approx 5.10$ ). We also know our hypothesized population mean from the null hypothesis is $\mu_0 = 90.8$ , and our sample size is $n = 5$ . Now we have all the pieces we need to calculate the t-statistic!

Remember the formula?

$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

Let's plug in the values:

$t = \frac{83.52 - 90.8}{5.10 / \sqrt{5}} = \frac{-7.28}{5.10 / 2.236} = \frac{-7.28}{2.281} \approx -3.19$

So, our calculated t-statistic is approximately -3.19. This value tells us how many standard errors our sample mean is away from the hypothesized population mean. Since it's negative, it means our sample mean is lower than the hypothesized mean, which aligns with our alternative hypothesis ( $H_a: \mu < 90.8$ ).

Now, what do we do with this t-statistic? We need to compare it to a critical value (or calculate a p-value) to determine whether we should reject the null hypothesis. Let's move on to the next step to figure that out!

Step 4: Determine the Critical Value (or P-value)

Okay, guys, we've got our t-statistic ( $t \approx -3.19$ ). Now it's time to figure out if this is extreme enough to reject the null hypothesis. We have two main ways to do this: the critical value approach and the p-value approach. Let's explore both!

Critical Value Approach

The critical value approach involves comparing our calculated t-statistic to a critical value from the t-distribution. The critical value is a threshold that tells us how extreme our test statistic needs to be to reject the null hypothesis.

Here's how we find the critical value:

Determine the degrees of freedom (df): The degrees of freedom for a t-test are calculated as $df = n - 1$ . In our case, $n = 5$ , so $df = 5 - 1 = 4$ .
Determine the significance level ( $\alpha$ ): We were given $\alpha = 0.002$ . This is the probability of rejecting the null hypothesis when it's actually true (a Type I error).
Determine the type of test: Our alternative hypothesis is $H_a: \mu < 90.8$ , which is a left-tailed test. This means we're only interested in extreme values in the left tail of the t-distribution.
Look up the critical value in a t-table: We need to find the t-value in a t-table with 4 degrees of freedom and a significance level of 0.002 for a one-tailed test. T-tables can vary slightly, but you're looking for the value that corresponds to the area in the left tail being 0.002. You might have to interpolate if your table doesn't have the exact value. For this case the value should be around -4.604.

So, our critical value is approximately -4.604. Now we compare our calculated t-statistic (-3.19) to this critical value. If our t-statistic is less than (more negative than) the critical value, we reject the null hypothesis. In our case, -3.19 is not less than -4.604.

P-value Approach

The p-value approach is another way to decide whether to reject the null hypothesis. The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one we calculated, assuming the null hypothesis is true.

Here's how we find the p-value:

Use a t-distribution calculator or software: You can use a t-distribution calculator online, statistical software (like R or SPSS), or a calculator with statistical functions to find the p-value.
Input the t-statistic, degrees of freedom, and type of test: We have $t = -3.19$ , $df = 4$ , and it's a left-tailed test.
Calculate the p-value: The calculator will give you the area in the left tail of the t-distribution corresponding to our t-statistic. For this example the p-value will be approximately 0.0167.

So, our p-value is approximately 0.0167. Now we compare this p-value to our significance level ( $\alpha = 0.002$ ). If the p-value is less than $\alpha$ , we reject the null hypothesis. In our case, 0.0167 is not less than 0.002.

Step 5: Make a Decision and Conclusion

Alright, guys, we've done the calculations, we've compared our results to critical values and p-values. Now it's time for the grand finale: making a decision about the null hypothesis and drawing a conclusion!

Decision

Using the critical value approach: We found our calculated t-statistic (-3.19) was not less than our critical value (-4.604). Therefore, we do not reject the null hypothesis.
Using the p-value approach: We found our p-value (0.0167) was not less than our significance level (0.002). Therefore, we do not reject the null hypothesis.

Both approaches lead us to the same decision: we fail to reject the null hypothesis. This means we don't have enough evidence to support the alternative hypothesis.

Conclusion

Now, let's put this into plain English. What does it mean that we failed to reject the null hypothesis?

In the context of our problem, it means that based on our sample data, we do not have sufficient evidence to conclude that the population mean ( $\mu$ ) is less than 90.8 at a significance level of 0.002.

It's important to be clear about what this doesn't mean. It doesn't mean we've proven that the population mean is 90.8. It just means we haven't found enough evidence to say it's less than 90.8. There's a subtle but important difference!

Think of it like a court case. The null hypothesis is like the presumption of innocence. We need enough evidence to convict (reject the null hypothesis). If we don't have enough evidence, we don't declare the person guilty (reject the null hypothesis), but that doesn't mean we've proven they're innocent (that the null hypothesis is true).

So, there you have it! We've walked through a complete hypothesis test for a population mean when the population standard deviation is unknown. You guys rocked it! Remember the steps, practice with different examples, and you'll be hypothesis testing like a pro in no time. Keep those statistical gears turning, and we'll catch you in the next article!