Technology & Engineering
Hypothesis Test for a Population Mean
A hypothesis test for a population mean is a statistical method used to make inferences about the true mean of a population based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, collecting sample data, calculating a test statistic, and determining the likelihood of observing the sample mean if the null hypothesis is true. This test helps to assess whether there is enough evidence to support a claim about the population mean.
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8 Key excerpts on "Hypothesis Test for a Population Mean"
- Mikel J. Harry, Prem S. Mann, Ofelia C. De Hodgins, Richard L. Hulbert, Christopher J. Lacke(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
16 Single-Population Hypothesis Tests 16.1 OVERVIEW This chapter introduces the second topic in inferential statistics. tests of hypotheses. In a hypothesis test, we test a certain given theory or belief about a population parameter. We may want to find out, using some sample information, whether or not a given claim (or statement) about a population parameter is reasonable. This chapter discusses how to make such tests of hypotheses about the population mean µ and the population proportion p. As an example, a soft-drink company may claim that, on average, its cans contain 12 ounces (oz) of soda. A government agency may want to test whether or not such cans contain, on average, 12 oz of soda. As another example, according to the US Bureau of Labor Statistics, 57.3% of married women in the United States were working outside their homes in 1991. An economist may want to check if this percentage is still reasonable for this year. In the first of these two examples we are to test a hypothesis about the population mean µ, and in the second example we are to test a hypothesis about the population proportion p. 16.2 INTRODUCTION TO HYPOTHESIS TESTING Why do we need to perform a hypothesis test? Reconsider the example about soft-drink cans. Suppose that we take a sample of 100 cans of the soft drink under investigation. We then find out that the mean amount of soda in these 100 cans is 11.89 oz. On the basis of this result, can we state that, on average, all such cans contain less than 12 oz of soda and that the company is lying to the public? Not until we perform a hypothesis test can we make such an accusation. The reason is that the mean = 11. 89 oz is obtained from a sample. The difference between 12 oz (the required average for the population) and 11.89 oz (the observed average for the sample) may have occurred only because of the variability from can to can. Another sample of 100 cans may give us a mean of 12.04 oz- eBook - ePub
- Derek L. Waller(Author)
- 2016(Publication Date)
- Routledge(Publisher)
research hypothesis since it is a value that has been obtained from a sampling experiment.For example, the hypothesis is that the average age of the population in a certain country is 35. This value is the null hypothesis. The alternative to the null hypothesis is that the average age of the population is not 35 but is some other value. In hypothesis testing there are three possibilities. The first is that there is evidence that the value is significantly different from the hypothesized value. The second is that there is evidence that the value is significantly greater than the hypothesized value. The third is that there is evidence that the value is significantly less than the hypothesized value. Note that in these sentences we say there is evidence because as always in statistics there is no guarantee of the result as we are basing our analysis of the population on sampling alone and of course our sample experiment may not yield the correct result. These three possibilities lead to using a two-tailed hypothesis test , a right-tailed hypothesis test , and a left-tailed hypothesis test as explained in the next section.Presenting hypothesis testing for the mean
In hypothesis testing for the mean an assumption is made about the mean or average value of the population. Then we take a sample from this population, determine the sample mean value, and measure the difference between this sample mean and the hypothesized population value. If the difference between the sample mean and the hypothesized population mean is small, then the probability is higher that our hypothesized population mean value is correct. If the difference is large then the probability is lower that our hypothesized value is correct.Two-tailed test
A two-tailed test is used when we are testing to see if a value is significantly different - eBook - ePub
- Deborah J. Rumsey(Author)
- 2019(Publication Date)
- For Dummies(Publisher)
Alpha levels and resulting decisions will vary from researcher to researcher. General steps for a hypothesis test Here’s a boiled-down summary of the steps involved in doing a hypothesis test. (Particular formulas needed to find test statistics for any of the most common hypothesis tests are provided in the rest of this chapter.) Set up the null and alternative hypotheses: and. Take a random sample of individuals from the population and calculate the sample statistics (means and standard deviations). Convert the sample statistic to a test statistic by changing it to a standard score (all formulas for test statistics are provided later in this chapter). Find the p -value for your test statistic. Examine your p -value and make your decision. Testing One Population Mean This test is used when the variable is numerical and only one population or group is being studied. For example, Dr. Phil says that the average time that working mothers spend talking to their children is 11 minutes per day. The variable, time, is numerical, and the population is all working mothers. The null hypothesis in the Dr. Phil example is minutes. Note that represents the average number of minutes per day that all working mothers spend talking to their children, and the claim is that that mean is 11. The alternative hypothesis,, is either:, or. Let’s suppose you suspect that the average time working mothers spend talking with their kids is more than 11 minutes; your alternative hypothesis would be. The formula for the test statistic for one population mean is. To calculate it, do the following: Calculate the sample mean,, and the sample standard deviation, s. Let n represent the sample size. See Chapter 1 for calculations of the mean and standard deviation. Find minus. (Remember, is the claimed value of the population mean.) Calculate the standard error:. Divide your result from Step 2 by the standard error found in Step 3. For the Dr - eBook - ePub
Business Statistics Using Excel
A Complete Course in Data Analytics
- R. Panneerselvam(Author)
- 2023(Publication Date)
- Routledge India(Publisher)
10 Testing HypothesesDOI: 10.4324/9781032671703-10Learning ObjectivesThe study of this chapter will enable the readers to- Understand the concept of a hypothesis and its importance in sampling.
- Analyse real-life problems through tests concerning a single mean when the mean and variance of the populations are known and the size of the population is finite using Z sampling statistics.
- Understand the process of testing hypotheses concerning the difference between two means when the variances of the populations are known and the sample sizes are large using Z sampling statistics.
- Study the tests of hypotheses concerning a single mean when the variance of the population is unknown and the sample size is small using t tests.
- Analyse real-life problems through tests concerning the difference between two means when the variances of the population are unknown and equal and the sample size is small using t tests.
- Analyse real-life problems through tests concerning the difference between two means when the variances of the population are unknown and unequal and the sample size is small using t tests.
- Study testing hypotheses of real-life problems using the paired t test.
10.1 Introduction
A hypothesis an assumption about a population. The two categories of hypotheses are the null hypothesis and alternate hypothesis. In the null hypothesis(, an assumption about the population will be made, like the sample mean is less than a specified constant. The alternate hypothesis (H1 :H o:X ¯≤ k )X ¯≤ > k) is opposite the null hypothesis, say, the sample mean is more than the specified constant. In data analysis, three types of hypothesis testing (≤, ≥, and =) will be carried out. In reality, based on the mean(s) and variance(s) of the sample(s) collected from the population, the investigator has to take a decision to accept or reject the null hypothesis. In this process, a significance level (α - Bilal M. Ayyub, Richard H. McCuen(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
The null hypothesis is rejected when the computed value lies in the region of rejection. Rejection of the null hypothesis implies acceptance of the alternative hypothesis. When a computed value of the test statistic lies in the region of rejection, there are two possible explanations. First, the sampling procedure may have resulted in an extreme value purely by chance; although this is a very unlikely event, this corresponds to the type I error of Table 9.1. Because the probability of this event is relatively small, this explanation is most often rejected. Second, the extreme value of the test statistic may have occurred because the null hypothesis is false; this is the explanation most often accepted and forms the basis for statistical inference. 9.3 HYPOTHESIS TESTS OF MEANS Because the mean of a population is often a primary element of statistical decision making, hypothesis tests of the mean are frequently used. The six-step procedure for testing hypotheses is applicable for testing hypotheses concerning the population mean μ. 9.3.1 Test of Mean with Known Population Variance When the standard deviation of the population is known, the procedure for testing the mean is as follows: Step 1: Formulate hypotheses —The null and alternative hypotheses must be stated in terms of the population parameter μ and the value selected for comparison, which may be denoted as μ 0. The three examples in Section 9.2.1 illustrate a test on a mean value. The null hypothesis should state that the mean of the population equals a preselected standard value. Acceptance of the null hypothesis implies that it is not significantly different from μ 0. Mathematically, the null hypothesis could be stated as H 0 : μ = μ 0 (9.3) The alternative hypothesis indicates that a difference exists- William P. Fox, Rodney X. Sturdivant(Authors)
- 2022(Publication Date)
- Chapman and Hall/CRC(Publisher)
10One Sample Hypothesis TestingDOI: 10.1201/9781003317906-10Objectives
- Know what a hypothesis test is.
- Be able to set up a hypothesis test.
- Be able to conduct the hypothesis test and reach a conclusion.
- Be able to use and interpret a p -value.
10.1 Introduction
You work for a researcher at a college and have collected data on the temperatures of students. The question you are trying to answer is whether or not the average temperature of an adult human being is still 98.6°F as was believed to be the case for many years. You have data on 1,000 students. The average temperature in the sample is 98.4°F, and the standard deviation is 0.6°F.Is there enough evidence to support the claim that the average human temperature has decreased? Proving something is hard to do, but hypothesis testing is the closest we get to proving or disproving a claim.A technique for inferring information about a parameter is a hypothesis test. A statistical hypothesis test is a claim about a single population characteristic or about values of several population characteristics. There is a null hypothesis (which is the claim initially favored or believed to be true, the “status quo”) and is denoted by H0 . The other hypothesis, the alternate hypothesis, which is often what the researcher hopes to show, is denoted as Ha . We will always keep equality with the null hypothesis. The objective is to decide, based upon sample information, which of the two claims is more plausible. Typical hypothesis tests can be categorized by three cases. Using the parameter of interest as the mean:- Case 1: H0 : μ = μ0 versus Ha : μ ≠ μ0
- Case 2: H0 : μ ≤ μ0 versus Ha : μ > μ0
- Case 3: H0 : μ ≥ μ0 versus Ha : μ < μ0
There are two types of errors that can be made in hypothesis testing, type I errors called α errors and type II errors called β- eBook - ePub
- David M. McEvoy(Author)
- 2018(Publication Date)
- Wiley(Publisher)
Chapter 8 Hypothesis Tests of a Population MeanIn 2010, a business professor at the University of Central Florida confronted the students in his strategic management course about cheating on a mid-term exam.1 The class was huge, with over 600 students. The professor was convinced that roughly 200 students had cheated by getting the answers to the exam in advance. The interesting part is that he had no direct evidence. The exams were proctored in a laboratory environment and not a single student was actively caught cheating. Rather, the professor relied on statistics to drive his initial suspicion. The professor had been teaching the class for many semesters and as part of the discovery process, he conducted hypothesis tests to compare that semester's exam grades with the historic grades from past semesters. He found a significant difference. The difference was so significant that he was confident that the abnormally high grades were not due to random chance. Rather, the difference in exam grades was large enough to strongly suspect foul play. His suspicions were later confirmed by a student who tipped the professor off about students accessing the exam questions online. The guilty students eventually admitted to cheating and the entire class was required to take a new exam. While statistical analysis was not the only thing used to confirm the suspicion of cheating, the results of hypothesis testing triggered further inquiry.The real power of statistics is being able to use samples of data to infer something unknown about a larger population. We explored one aspect of this in Chapter 7 on confidence intervals. Here, we move onto using sample data to test hypotheses regarding the larger population. We start with the simplest case in which we use one sample of data to test a single numeric value regarding the population. Some good examples are hypothesis tests for quality control of products and services. Starbucks, for example, reports that a 16 oz cup of regular coffee (the grande size of Pike Place) has 310 milligrams (mgs) of caffeine.2 Of course, there is going to be some variation in the actual amount of caffeine found in every grande cup of coffee brewed across the United States and abroad. The caffeine content can depend on the water quality, how long the beans have been roasted, the age of the beans, the skillset of the barista, the type of filter, and many other things. Therefore, we would not expect every single 16 oz cup to have exactly 310 mg of coffee. Given this variation, it is possible to sample a number of cups of coffee to test whether the average - eBook - ePub
- Richard A. DeFusco, Dennis W. McLeavey, Jerald E. Pinto, David E. Runkle, Mark J. P. Anson(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
- The statistical decision consists of rejecting or not rejecting the null hypothesis. The economic decision takes into consideration all economic issues pertinent to the decision.
- The p-value is the smallest level of significance at which the null hypothesis can be rejected. The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis. The p-value approach to hypothesis testing does not involve setting a significance level; rather it involves computing a p-value for the test statistic and allowing the consumer of the research to interpret its significance.
- For hypothesis tests concerning the population mean of a normally distributed population with unknown (known) variance, the theoretically correct test statistic is the t-statistic (z-statistic). In the unknown variance case, given large samples (generally, samples of 30 or more observations), the z-statistic may be used in place of the t-statistic because of the force of the central limit theorem.
- The t-distribution is a symmetrical distribution defined by a single parameter: degrees of freedom. Compared to the standard normal distribution, the t-distribution has fatter tails.
- When we want to test whether the observed difference between two means is statistically significant, we must first decide whether the samples are independent or dependent (related). If the samples are independent, we conduct tests concerning differences between means. If the samples are dependent, we conduct tests of mean differences (paired comparisons tests).
- When we conduct a test of the difference between two population means from normally distributed populations with unknown variances, if we can assume the variances are equal, we use a t
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