Hypothesis Testing Using Rejection Region(s) In Exercises 39–4...

Question

Hypothesis Testing Using Rejection Region(s) In Exercises 39–44, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

[APPLET] Fluorescent Lamps A compact fluorescent lamp (CFL) bulb manufacturer guarantees that the mean life of a CFL bulb is at least 10,000 hours. You want to test this guarantee. To do so, you record the lives of a random sample of 32 CFL bulbs. The results (in hours) are listed. Assume the population standard deviation is 1850 hours. At alpha=0.11, do you have enough evidence to reject the manufacturer’s claim?

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Accepted Answer

Hello everyone. Let's take a look at this question together. A nutrition bar producer claims that the mean protein content in their bars is at least 20 g. A random sample of 50 bars has a mean protein content of 19.1 g. The population standard deviation is 2.7 g. At alpha equals 0.09, does the data provide enough evidence to reject the producer's claim? So in order to solve this question, we have to determine whether the data provides enough evidence to reject the producer's claim that the mean protein content in their bars is at least 20 g, where we have a random sample of 50 bars that has a mean protein content of 19.1 g, and the population standard deviation is 2.7 g. And so looking at the information provided in the question, we know that the sample size N is equal to 50, and using the provided information to conduct our requirement check, we know since the population standard deviation is known, the sample is random, and our sample size N is greater than 30, we can use the Z test. And the first step in solving this problem is to state the null and alternative hypotheses. And we know that the company claims that the average protein content in their bars is at least 20 g. Which this statement can be expressed as mu is greater than or equal to 20. And since it is a statement of equality, it is the null hypothesis. So our null hypothesis is, is greater than or equal to 20, and the complement of the null hypothesis is the alternative hypothesis. So our alternative hypothesis is mute is less than 20. Therefore, null and alternative hypotheses are the null hypothesis is mu is greater than or equal to 20, which is our claim, and the alternative hypothesis is mu is less than 20. And from the provided information, the given level of significance for alpha is alpha equals 0.09. Now we can determine the critical value of the sub-zero, and since the alternative hypothesis contains the less than inequality symbol, the hypothesis test is a left-tailed test. So we will use the table for standard normal distribution and look for the Z score that corresponds to a cumulative area. is to the left of 0.09. And since 0.09 is between 0.0885 and 0.0901, we can interpolate. So by interpolation, we are given the following equation, where we know that we have to solve for the value of Z sub-zero. So we rewrite the equation to solve for Z subzero, and when simplified, our critical value of Z subzero is equal to -1.341. And using our critical value, we can determine the rejection region. And since the test is a left tailed test, and the critical value is -1.341, the rejection region is Z is less than -1.341. And now we have to find the standardized test. Statistic of Z using the formula Z is equal to the sample mean subtracted by the population mean, which is divided by the population standard deviation divided by the square root of our sample size. And substituting the values into the formula, we have 19.1, subtracted by 20, which is divided by 2.7 divided by the square root of 50. And simplifying this equation, we have -2.357 as the value for the standardized test statistic Z. Which we will use this value and our critical value to make a decision, and since the value of Z is less than our critical value in a left tailed test, Z is in the rejection region. So we will reject the null hypothesis. And since the claim is the null hypothesis, and we reject the null hypothesis, the interpretation for this decision is there is sufficient evidence to reject the claim that the mean protein content in the bars is at least 20 g. Therefore, based on our answer choices, we cannot. Identify that answer choice C is the correct answer. As we know that there is sufficient evidence to reject the claim that the mean protein content in the bars is at least 20 g, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Hypothesis Testing

Rejection Region

Standardized Test Statistic (z)

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