Hypothesis testing using critical values

Fat in ground meat (Hypothesis testing for the mean)

I like ground meat recently. But I’m afraid of gaining weight, so I usually buy the 200-gram package with only 3% fat. Yet I feel like I’m gaining more weight than I expected. When I stand by my boyfriend, who is tall and thin, I feel like we are making the number 10. So I started to distrust the “3% of fat” that the producer claims on the labels. So I bought a magic machine that can detect the amount of fat in a package of meat, and I want to test if it’s true that the ground meat package on average has only “3% fat.”

I randomly bought 35 packages of meat from local stores and brought them home to scan. After calculating, I found that the sample mean of fat in each package is $\bar{x} = 6.8$ (grams) and the sample standard deviation is $s = 2$ grams.

The test statistic explained:

Recall from the previous section that if $X_1, X_2, \ldots, X_n$ are independent identically distributed with mean $\mu$ and variance $\sigma^2$ and let $\bar{X}$ be the sample mean, then as $n \to \infty$

$\frac{\bar{X} - \mu}{\sigma / \sqrt{n}} = \frac{n(\bar{X} - \mu)}{\sigma} \to N(0,1)$

This is the theoretical ground for the z-test for the mean that we will use for our test procedure.

The test procedure:

Now let’s conduct a hypothesis testing procedure.

Here the sample size is $n = 35$ , which is large enough to say the amount of fat in the packages follows a normal distribution. Therefore, I’ll conduct the z-test for the mean (We usually use the z-test when the sample size is at least 30).

First, to set the null and alternate hypothesis, note that 3% of 200 grams is 6 grams. Therefore my null hypothesis is

$H_0: \mu = 6$

because I want to reject the fact that on average the packages contain 3% of fat. I think the packages contain more than 3% on average. Therefore my alternative hypothesis is

$H_A: \mu > 6$

For me, I will choose a significance level $\alpha$ of 0.05.

The test statistic is

$Z = \frac{\bar{x} - 6}{s / \sqrt{n}} = \frac{6.8 - 6}{2 / \sqrt{35}} = \frac{0.8 \times \sqrt{35}}{2} \approx 2.37$

In the figure above, the orange dot region is the critical region (rejection region), which contains the values that should not happen if the null hypothesis is true. The critical value is the value that separates the normal region and the rejection region. Here by the normal region (acceptance region) I mean the region where $\bar{x}$ should fall into if $H_0$ is true. The area of the orange dot region is the significant value $\alpha$ (Note that the area under the curve is 1, just as the probabilities of all events should sum to 1. The right tail part (the orange dot region) represents the observations that are bigger than expected, i.e., abnormal region). Here we choose $\alpha = 0.05$ which leads to the critical value $z_{0.05} = 1.96$ . So we consider all values that are bigger than $z_{0.05} = 1.96$ to be abnormal. Note that

$Z \approx 2.37 > z_{0.05}$

Therefore we reject the null hypothesis and conclude that on average each package of ground meat contains more than 3% fat. Oh my God, that’s why I’m gaining so much weight! Yet I don’t want to waste my food and I’ll eat the meat packages I bought. My boyfriend and I will probably be number 10 for longer than I expected, but 10 is a nice number anyway. Also, I’ll email the producer about my findings to ask for a refund for all the packages that I purchased or… give me a coupon for a free year of their meat… I mean their meat products.

Summary:

The process of hypothesis testing using a critical value is as follows:

Formulate hypotheses:

Null Hypothesis ( $H_0$ ): Represents the default or status quo hypothesis, often stating no effect or no difference.
Alternative Hypothesis ( $H_A$ ): Represents the hypothesis you want to test, often suggesting an effect or a difference.

Choose a significance level (alpha, often set at 0.05): This is the probability threshold used to make decisions about the null hypothesis.
Select a statistical test and calculate the test statistic based on your sample data.
Determine the acceptance region: The acceptance region is the range of values for the test statistic that, if observed, leads to the acceptance of the null hypothesis. These values are determined based on the chosen significance level and the distribution of the test statistic under the null hypothesis.
Compare the calculated test statistic to the critical value(s): If the test statistic falls within the acceptance region (i.e., it does not exceed the critical value), you fail to reject the null hypothesis. If the test statistic falls outside the acceptance region (i.e., it exceeds the critical value), you reject the null hypothesis in favor of the alternative hypothesis.

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Hypothesis testing using critical values

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