One-mean t-test (Hypothesis test)

Question: Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

As with all hypothesis test problems, we're going to take the following steps to solve!

  • Step 1 - State the hypothesis
  • Step 2 - Calculate the test statistic
  • Step 3 - Find the p-value
  • Step 4 - Make the concluding statement

Step 1 - State the hypotheses

Let's start with the null hypothesis (H0), then we'll move onto the alternative hypothesis (Ha).

Defining your null hypothesis (H0)

Here, we need to ask ourselves: what is the baseline claim that's being made about the population?

In the case of the situation above, it's that Sigma Apple Pi brothers get an average of 25 Tinder matches per day.

Question: Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

We'd write that null hypothesis (H0) like so:

H0: µ = 25

Essentially, what we're saying here is that the true population mean for the average Tinder matches per day of Sigma Apple Pi brothers...

H0: µ = 25

...is equal to 25.

H0: µ = 25

Defining your alternative hypothesis (Ha)

Here we need to ask ourselves: what's the claim that's being made that's different from the baseline claim about the population?

In the case of the situation above, we're claiming that Sigma Apple Pi brothers actually get less than 25 average Tinder matches per day.

Question: Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

We'd write that alternative hypothesis (Ha) like so:

H0: µ = 25
Ha: µ < 25

Essentially what we're saying here is that the true population mean for average Tinder matches per day of Sigma Apple Pi brothers...

H0: µ = 25
Ha: µ < 25

...is actually less than 25.

H0: µ = 25
Ha: µ < 25

Step 2 - Calculating your test statistic

Before even beginning to calculate our test statistic, we have to check out assumptions!

Check your assumptions!

We're working with a sample mean here, so in accordance with Assumptions for sampling distributions, that means we must check the following assumptions:

1. Sample is randomly selected from the population
2. The sample size (n) is less than or equal to 10% of of the population size (N)
3. The sample size is greater than or equal to 30, or the population itself is normally distributed

Concerning #1, we're collecting a random sample from the population:

Question: Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

Concerning #2, we can assume that 35 freshmen students is less than 10% of the entire Sigma Apple Pi chapter at Crammer Nation University. (Fraternities at Crammer Nation University run big, we're talking 400+ members.)

Question: Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

Concerning #3, our sample size is greater than 30!

Question: Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

Therefore, all of our assumptions are passed! Now we can move onto calculating our test statistic!

Which should we use: z-score or t-score?

Based on this graphic in t-scores and sampling distributions...

...since we're given the sample standard deviation (s) in the prompt and not the population standard deviation (σ)...

Question: You're going through fraternity rush at Crammer Nation University and hear that Sigma Apple Pi brothers get a lot of Tinder matches. You take a random sample of 35 brothers and find a sample mean of 23.2 daily Tinder matches with a standard deviation of 3.2 matches. Based on this, find a 95% confidence interval for the true mean daily Tinder matches of Sigma Apple Pi brothers.

...we'll be calculate a t-score here! We'll utilize the following formula to calculate that t-score:

What if we had the population standard deviation (σ)?

Then since our sample size is above 30, according to the Central Limit Theorem we'd be able to use the following formula to calculate the z-score instead of t-score!

Notice the only difference here are that you're plugging in the population standard deviation (σ) instead of the sample standard deviation (s)!

This looks similar to the regular t-score formula... what's with the "µ0"?

You'll notice that this formula is extremely similar to the t-score formula in What is a t-score?...

...but now it's µ0 instead of µ.

Long story short, that's because with hypothesis tests, we use the µ0 (that little "0" is often called "knot") to signify that we don't know for sure that it's the population mean. It's the claimed population mean that's being tested within our hypothesis test!

Plugging in sample mean (x-bar)

Based on the prompt, the sample mean is 23.5 daily Tinder matches...

Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

...therefore, we'll plug in 23.5 for x-bar.

Plugging in claimed population mean (µ0)

The claimed population mean of Sigma Apple Pi daily Tinder matches is 25...

Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

...therefore, we'll plug in 25 for µ0.

Plugging in sample standard deviation (s)

The sample standard deviation is 5.7 matches...

Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

...so we'll plug in 5.7 for s.

Plugging in sample size (n)

Lastly, our sample size is 35 brothers...

Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

...therefore, we'll plug in 35 for n.

Solving for t-score

When we solve this out, it results in a t-score of -1.558!

Step 3 - Find your p-value

First, let's determine if we're doing a one-tail or two-tail test, and then let's find our p-value in the z-table.

One-tail or two-tail?

In What is a hypothesis test?, we declared the following:

If your alternative hypothesis has ">" or "<", that means it's a one-tail test.
If your alternative hypothesis has "", that means it's a two-tail test.

Since our alternative hypothesis uses a "<", that means we're dealing with a one-tail test, specifically a left-tail test, since we're using the less than symbol.

At an alpha level of 0.05...

Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

...that means we'll be looking to see if our p-value occurs in this area under our sampling distribution curve.

Finding our p-value

We know our t-score is -1.558, so that means we'll be using the t-table to find a range of p-values.

To find that range, we first need to determine our degrees of freedom (df).

Calculating our degrees of freedom (df)

We can calculate our degrees of freedom (df) easily with the below formula:

df = n - 1

In our case, we've got a sample size of 35...

df = 35 - 1

...therefore our degrees of freedom are 34.

df = 34

Locating our t-score range

Now that we've got our degrees of freedom (df) of 34, let's go to the row in the t-table corresponding to 34 degrees of freedom.

Now ask yourself: what range of t-scores (within the row for 34 degrees of freedom) does our t-score of -1.558 (ignore the negative) fall between?

It falls between 1.307 and 1.691!

And, what p-values do these t-scores correspond to?

0.10 and 0.05!

Therefore, we know that the p-value for our t-score of -1.558 falls somewhere between 0.10 and 0.05!

0.05 < p < 0.10

Visualizing our p-value

The graphic at the top of the t-table shows the following...

...meaning that all the p-values in it represent the area to the right of each t-score.

However, since our t-score is negative, it's practically like flipping the graphic...

...meaning that the p-values in the t-table correctly correspond to the area to the left of each negative t-score. This is exactly what we are looking for, since we're working with the left-tail of our sampling distribution!

Based on the t-table, we found that our t-score of -1.558...

...falls somewhere in the range of a p-value of 0.05 and 0.10.

And we know that all values between 0.05 and 0.10 are going to fall outside our alpha value (α) of 0.05!

Why do we not need an exact p-value?

Because in hypothesis tests, all that matters is whether or not your p-value is above or below your alpha level!

Knowing that our p-value lies somewhere between 0.05 and 0.10...

0.05 < p < 0.10

...is enough intel for us to determine that our actual p-value corresponding to a t-score of -1.558 is above our alpha level!

Step 4 - Make your concluding statement

Your concluding statement is going to center around the alpha level declared in the problem. In most cases, that alpha level will be 0.05. Each problem should explicitly state the alpha level. In our problem, it's 0.05.

Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You have a hunch that their daily Tinder matches are actually lower than that, so you collect a random sample of 35 Sigma Apple Pi brothers' daily Tinder matches. You measure a mean of 23.5 daily matches with a standard deviation of 5.7 matches. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

As we stated in What is a hypothesis test?...

- If the p-value is below the alpha level, then we reject the null hypothesis
- If the p-value is above the alpha level, then we fail to reject the null hypothesis.

In our case, our p-value range above the alpha level. Remember: all values between 0.05 and 0.10 are greater than 0.05!

0.05 < p < 0.10

This means that we fail to reject our null hypothesis!

Applying the t-score answer template

If you remember in What is a hypothesis test?, we gave the following answer template when working with t-scores:

Since our p-value range of p-value range is less / greater than our alpha level of alpha level value, we reject / fail to reject the null hypothesis. We do / don't have enough evidence to support the alternative hypothesis, which states that description of the alternative hypothesis.

Applied to our question, this would give us the following answer to our original question!

Answer: Since our p-value range of 0.05 < p < 0.10 is greater than our alpha level of 0.05, we fail to reject the null hypothesis. We don't have enough evidence to support the alternative hypothesis, which states that the brothers of Sigma Apple Pi have an mean daily Tinder match value less than 25.

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