9. Hypothesis Testing for One Sample

Critical Values and Rejection Regions

9. Hypothesis Testing for One Sample

Critical Values and Rejection Regions

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Multiple Choice

A soup company claims that the average sodium content of their most popular soup is 500 mg per can. A nutritionist collects a sample of 36 cans with mean sodium content 507 mg. Assume a known pop. standard deviation of 15 mg & test the nutritionist’s suspicion that the mean sodium content is more than 500 mg using the critical value method with $alpha equals 0.01$ .

Critical Value(s): $2.8$

Because test stat. $(z)$ is INSIDE rejection region, we REJECT $H sub 0$ . There is ENOUGH evidence to conclude that $mu is greater than 500$

Critical Value(s): $2.8$

Because test stat. $(z)$ is OUTSIDE rejection region, we FAIL TO REJECT $H sub 0$ . There is NOT ENOUGH evidence to conclude that $mu is greater than 500$

Critical Value(s): $2.33$

Because test stat. $(z)$ is INSIDE rejection region, we REJECT $H sub 0$ . There is ENOUGH evidence to conclude that $mu is greater than 500$

Critical Value(s): $2.33$

Because test stat. $(z)$ is OUTSIDE rejection region, we FAIL TO REJECT $H sub 0$ . There is NOT ENOUGH evidence to conclude that $mu is greater than 500$

Verified step by step guidance

Step 1: Define the null and alternative hypotheses. Since the nutritionist suspects the mean sodium content is more than 500 mg, set up the hypotheses as: $H_0: \mu = 500$ (null hypothesis) $H_a: \mu > 500$ (alternative hypothesis, one-tailed test).

Step 2: Identify the given values from the problem: Sample mean $\overline{x} = 507$ mg, population standard deviation $\sigma = 15$ mg, sample size $n = 36$, and significance level $\alpha = 0.01$.

Step 3: Calculate the test statistic using the z-test formula for a known population standard deviation: $z = \frac{\overline{x} - \mu}{\sigma / \sqrt{n}}$ Substitute the values to express the formula before calculating the numerical value.

Step 4: Determine the critical value for a right-tailed test at $\alpha = 0.01$. This critical z-value corresponds to the point beyond which we reject the null hypothesis. You can find this value using a z-table or statistical software.

Step 5: Compare the calculated test statistic $z$ to the critical value: - If $z$ is greater than the critical value, reject $H_0$. - If $z$ is less than or equal to the critical value, fail to reject $H_0$. Based on this comparison, conclude whether there is enough evidence to support the claim that the mean sodium content is greater than 500 mg.

6:25m

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