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.

Light Bulbs A light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 750 hours. A random sample of 25 light bulbs has a mean life of 745 hours. Assume the population is normally distributed and the population standard deviation is 60 hours. At alpha= 0.02, do you have enough evidence to reject the manufacturer’s claim?

Accepted Answer

Hello, everyone. Let's take a look at this question together. A battery company claims that the mean lifespan of its batteries is at least 1200 hours. A random sample of 36 batteries has a mean lifespan of 1,185 hours. The population standard deviation is known to be 72 hours. At alpha equals 0.01, is there sufficient evidence to reject the company's claim. So in order to solve this question, we have to determine whether there is sufficient evidence to reject the company's claim that the mean lifespan of its batteries is at least 1200 hours if we have a random sample of 36 batteries with a mean lifespan of 1,185 hours, and the population standard deviation is known to be 72. Hours. And so using the information provided in the question, we should note that the sample size is n equals 36, and 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. We can use the Z test. And so the first step in solving this problem is stating the and alternative hypotheses, which we know that the company claims that the mean lifespan of its batteries is at least 1200 hours, and this can be expressed as mu is greater than or equal to 1200. And since it is a statement of equality, it is the null hypothesis. So our null hypothesis is mu is greater than or equal. To 1200, and we know the complement of the null hypothesis is the alternative hypothesis, so our alternative hypothesis is mu is less than 1200. Therefore, we have our null and alternative hypotheses, which are the null hypothesis is mu is greater than or equal to 1200, which is the claim, and our alternative hypothesis. is less than 1200. And looking at the information provided, our given level of significance, which is alpha, is alpha equals 0.01. And now we have to determine the critical value which is the sub-zero, and since our 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, which is to the left of 0.01. And since 0.01 is between 0.0099 and 0.0102, we can interpolate, and by interpolation, we have the following equation, which we can then rewrite this equation to solve for Z subzero, which is then rewritten as the following where we solve for the value for Z subzero, which is our critical. So by interpolation, the critical value of Z subzero is equal to -2.327. And then we determine the rejection region, and since the test is a left tail test and the critical value is Z subzero is equal to -2.327, the rejection region is Z is less than -2.327. And then we find the standardized test statistic, which is the value 4 Z. And can be found using the following formula, where Z is equal to the sample mean subtracted by the population mean divided by the population standard deviation, which is divided by the square root of the sample size N. And substituting the values into the formula, we have 1,185, subtracted by 1200 divided by 72, which is divided by the square root of 36. So the value for the standardized test statistics, Z is -1.25. And using this data, we can make a decision, which we know since Z is greater than our critical value of Z subzero in a left tail test, Z is not in the rejection region. Therefore, we fail to reject the null hypothesis. And since the claim is the null hypothesis and we fail to reject the null hypothesis, the interpretation for this decision is that there is insufficient evidence to reject the claim that the mean life span of the batteries is at least 1200 hours and based on our interpret. we can identify that answer choice B is the correct answer, as there is insufficient evidence to reject the claim that the mean life span of the batteries is at least 1200 hours, 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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