Transcription
Good day everyone. Today we will discuss solving problems involving tests of hypotheses on population mean. Before this lesson, you studied about constructing hypotheses based on assumptions made. You'll learn how to determine the appropriate test statistic to be used and solve its value in a given situation, as well as how to identify the critical value and draw the critical region. So, in this video lesson, you will apply knowledge and skills on solving problems in hypothesis testing. Eventually, you will decide whether you will reject the null hypothesis or not.
After going through this lesson, you are expected to identify the steps in hypothesis testing and solve problems involving tests of hypotheses on the population mean. Before we proceed to our discussion, let us answer the following questions to recall the concepts of hypothesis testing. First, the sign of the alternative hypothesis in a left-tailed test is always equal, less than, not equal, or greater than. Alright, the correct answer is less than.
Next, if the researcher wishes to test the claim that the mean of the population is 75, the appropriate null hypothesis is: the mean population less than or equal to 75, the mean population is greater than or equal to 75, the mean population is not equal to 75, and the mean population is equal to 75. So, what is the answer? Yes, the correct answer is the mean population is equal to 75.
Another, what kind of hypothesis is illustrated: "The mean score of all grade 11 students is higher than 75"? One-tailed tests, two-tailed tests, null hypothesis, alternative hypothesis. Since the claim is higher than, the correct answer is alternative hypothesis.
Another, based on the Central Limit Theorem, when the sample size is extremely large and the variance is unknown, what is the statistical test to be used? T-tests, two-tailed test, one-tailed test. So, what test statistic do we use since the sample size is extremely large? C-tests. Okay, the correct answer is t-test.
Next, what appropriate tool is applicable if the population is normal, sample standard deviation is known, and the sample is less than 30? [Music] P-test, normal test, and Central Limit Theorem. Since the sample size is less than 30, so less than 30, therefore the test statistic is t-test. [Music]
Next, if a sample size is 16 selected from a normal population and the sample mean is 56 and the sample standard deviation is 12, what statistical test is applicable to be used? F-tests. [Music] Z-test or Q-test. Since the sample size is less than 30, t-test is applicable in this given problem.
Next, in using t-test for a population mean, we assume that the sample is selected randomly. The given statement is always true, always false, sometimes true, sometimes false. So, the correct answer is the given statement is always true.
I want you to read and analyze the problem. According to survey, the average daily usage of social media worldwide of global internet users amounts to 142 minutes per day. Lance conducts his own survey among his friends to find out if their time spent on social media is significantly higher than the global survey. Do you think his claim is acceptable at five percent level of significance? So, in the table shows Lance's friends and their respective time spent on social social media. So, what is the process to answer the claim of Lance if it is acceptable or not? So, we can apply hypothesis testing.
In testing hypotheses on the population means, we follow the following steps. So, number one, formulate the null and alternative hypothesis. Number two, determine the level of significance in the hypothesis test. Number three, compute the test statistic. Number four, find the critical value and determine the rejection region. And last, decide and draw a conclusion based on the comparison of the calculated value of the test statistic and the critical value of the test.
Okay, let us go back to the given problem. So, first, let us identify the given data. Now, based on the given problem, what are the given data? So, the population mean is 142 minutes. Now, okay, 142 minutes and the sample standard deviation is 19.855. So, you can use your calculator here to solve the sample standard deviation using the given data. And then the sample mean is 152. So, using your calculator, you can add all the minutes per day, the corresponding minutes per day of Lance's friends and divide by 10 since this is 10 and the sample size is 10.
So, using the given data, let us apply the five steps in hypothesis testing. Okay, so again, the given data. And next is formulate the null and alternative hypothesis. Since the claim here, the claim here is higher than the global survey, so the alternative hypothesis will be: "The average online usage of his friends is higher than the global uses," which is translated into. Okay, your alternative hypothesis will be the population mean is greater than 142. So, based on the claim, can say higher than. So, if your alternative hypothesis is greater than, so what will be our null hypothesis? So, the null hypothesis is the same as or which it can be translated into an equal sign. Okay, from greater than, so your null hypothesis will be equal. Okay, or your population mean is equal to 142.
So, after formulating the null and alternative hypothesis, next is determine the level of significance, and that is 0.05 since 5% level of significance. And after that, identify the hypothesis test. Since our alternative hypothesis contains greater than, and that is a right-tailed test. So, after the after identifying the hypothesis test, we can now proceed in computing the test statistics. So, what applicable test statistic to be used in the given problems? So, based on our given, the sample size is less than 30. 10 is less than 30, therefore that is t-test. So, using the formula, we can compute now the t-value. So, what you're going to do is substitute all the given data in our formula. So, you can use your calculator. So, here, so you can input all the data. So, 152 minus 142, and then click the division bar, 19.855 over the square root of 10. So, the answer will be 1.593. So, so let's check that is correct. So, 1.593.
So, next is, let us now find the critical value by using the t-critical value. Since the sample size is 10, so therefore the degrees of freedom, the degrees of freedom is nine. Okay, so using the t-critical value and it is a one-tailed test, one-tailed test, the level of significance is 0.05, the degrees of freedom is 9. So, that is 1.833. Meaning the t-critical value is 1.833. And after identifying the critical value, let us now determine the rejection region. So, using the uh math generator or the graph generator, this is a free software in your browser. You can search imathas.com. Okay, so we can now determine the rejection region. So, what you're going to do is first, so we can select normal distribution and t-distribution rather, since this is t-test. And then you can encode the sample size, so that is 10, and choose right tail. So, choose right tail. The critical value is 1.833, and then the t-test value or the t-computed value is 1.593. And then you can uh click the. So, you can see the graph. So, as you can see, class, the red line, that is your computed value, and the shaded part of the graph is the critical region. So, as you can see, your computed value is at the acceptance region. So, going back to our uh presentation. So, here, that is your critical value. So, we all know your critical value is the a boundary between the rejection region and the non-rejection region. So, since your graph or your computed value is at the non-rejection region, so we can say that your computed value is at the non-rejection region.
So, what will be our decision? See, it since it is a right test and the t-computed value is 1.593 and it lies within the non-rejection region, so we failed to reject the null hypothesis. So, we can conclude there is no enough evidence to support the claim that the average online usage of his friends is higher than the global usage. Meaning the average online usage of his friends is the same as the global usage.
Let us now recall the important notes by answering the given questions. Let's start. Number one, if the t-computed value is 2.430 and the critical value is 2.011, what will be the decision? Letter A, reject the null hypothesis. Letter B, support the null hypothesis. Letter C, reject the alternative hypothesis. And letter D, do not reject the null hypothesis. The correct answer is letter A.
Next question. Number two, in the hypothesis testing procedure, drawing conclusion should always be the blank step. A, first. B, second. C, third. D, last. So, the correct answer is letter D, last step.
Number three, in a left-tailed test, what will you do if the critical value is less than the computed value? A, reject the null hypothesis. B, support the null hypothesis. Letter C, reject the alternative hypothesis. And letter D, do not reject the null hypothesis. The correct answer is letter A, reject the null hypothesis.
Number four, a one-sample t-test is conducted on the null hypothesis, which is the population mean 81.6. The sample has a mean of 84.1, the sample standard deviation is 3.1, with a sample size of 25, and the level of significance is 0.01. So, what conclusion can be drawn? Letter A, reject the alternative hypothesis. B, reject the null hypothesis. Letter C, failed to reject the null hypothesis. Letter D, fail to reject the alternative hypothesis. The correct answer is letter B, reject the null hypothesis.
And last, perform a hypothesis test where the null hypothesis is that your mean population is 6.9. A random sample of 16 items is selected. The sample mean is 7.1, and the sample standard deviation is 2.4. It can be assumed that the population is normally distributed at the level of significance of 5%. A, there is enough evidence to reject the claim. B, there is enough evidence to support the claim. Letter C, there is not enough evidence to reject the claim. And letter D, there is not enough evidence to support the claim. So, the correct answer is letter C, there is not enough evidence to reject the claim.