{"id":335,"date":"2016-04-21T22:43:40","date_gmt":"2016-04-21T22:43:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstats1xmaster\/?post_type=chapter&p=335"},"modified":"2022-01-02T22:25:55","modified_gmt":"2022-01-02T22:25:55","slug":"null-and-alternative-hypotheses","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/chapter\/null-and-alternative-hypotheses\/","title":{"raw":"Basics of Hypothesis Testing","rendered":"Basics of Hypothesis Testing"},"content":{"raw":"
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Learning Outcomes<\/h3>\r\n
\r\n
    \r\n \t
  • Describe hypothesis testing in general and in practice<\/li>\r\n \t
  • Differentiate between Type I and Type II Errors<\/li>\r\n \t
  • Conduct and interpret hypothesis tests for a single population mean, population standard deviation known<\/li>\r\n \t
  • Conduct and interpret hypothesis tests for a single population mean, population standard deviation unknown<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\nThe actual test begins by considering two\u00a0hypotheses<\/strong>. They are called the null hypothesis<\/strong> and the alternative hypothesis<\/strong>. These hypotheses contain opposing viewpoints.\r\n\r\nH0<\/sub><\/em>: The null hypothesis:<\/strong> It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.\r\n\r\nHa<\/sub><\/em>: The alternative hypothesis<\/strong>:<\/strong> It is a claim about the population that is contradictory to H0<\/sub><\/em> and what we conclude when we reject H0<\/sub><\/em>.\r\n\r\nSince the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.\r\n\r\nAfter you have determined which hypothesis the sample supports, you make a decision. There are two options for a\u00a0decision<\/strong>. They are \"reject H0<\/sub><\/em>\" if the sample information favors the alternative hypothesis or \"do not reject H0<\/sub><\/em>\" or \"decline to reject H0<\/sub><\/em>\" or \"fail to reject H0<\/sub><\/em>\" if the sample information is insufficient to reject the null hypothesis.\r\n\r\nMathematical Symbols Used in\u00a0H0<\/sub><\/em> and Ha<\/sub><\/em>:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    H0<\/sub><\/em><\/th>\r\nHa<\/sub><\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    equal (=)<\/td>\r\nnot equal (\u2260)\r\nor<\/strong> greater than (>) or<\/strong> less than (<)<\/td>\r\n<\/tr>\r\n
    greater than or equal to (\u2265)<\/td>\r\nless than (<)<\/td>\r\n<\/tr>\r\n
    less than or equal to (\u2264)<\/td>\r\nmore than (>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n
    \r\n\r\n

    Note<\/h4>\r\nH0<\/sub><\/em> always has a symbol with an equal in it. Ha<\/sub><\/em> never has a symbol with an equal in it.<\/strong> The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.\r\n\r\nhttps:\/\/youtu.be\/5D1gV37bKXY\r\n
    \r\n

    Example<\/h3>\r\nH0<\/sub><\/em>: No more than 30% of the registered voters in Santa Clara County voted in the primary election. p<\/em> \u2264 30\r\n\r\nHa<\/sub><\/em>: More than 30% of the registered voters in Santa Clara County voted in the primary election. p<\/em> > 30\r\n\r\n<\/div>\r\n

    <\/h3>\r\n
    \r\n

    try it<\/h3>\r\nA medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.\r\n\r\n[reveal-answer q=\"200489\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"200489\"]\r\n\r\nH0<\/sub><\/em> : The drug reduces cholesterol by 25%. p<\/em> = 0.25\r\n\r\nHa<\/sub><\/em> : The drug does not reduce cholesterol by 25%. p<\/em> \u2260 0.25\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    <\/h3>\r\n
    \r\n

    Example<\/h3>\r\nWe want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:\r\n\r\nH0<\/sub><\/em>: \u03bc<\/em> = 2.0\r\n\r\nHa<\/sub><\/em>: \u03bc<\/em> \u2260 2.0\r\n\r\n<\/div>\r\n

    <\/h3>\r\n
    \r\n

    try it<\/h3>\r\nWe want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, \u2260, \u2265, <, \u2264, >) for the null and alternative hypotheses. H0<\/sub><\/em>: \u03bc<\/em> __ 66 Ha<\/sub><\/em>:\u03bc<\/em> __ 66\r\n\r\n[reveal-answer q=\"395823\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"395823\"]\r\n\r\nH0<\/sub><\/em> : \u03bc<\/em> = 66\r\n\r\nHa<\/sub><\/em> : \u03bc<\/em> \u2260 66\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    <\/h3>\r\n
    \r\n

    Example<\/h3>\r\nWe want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:\r\n\r\nH0<\/sub><\/em>: \u03bc<\/em> \u2265 5\r\n\r\nHa<\/sub><\/em>: \u03bc<\/em> < 5\r\n\r\n<\/div>\r\n

    <\/h3>\r\n
    \r\n

    try it<\/h3>\r\nWe want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, \u2260, \u2265, <, \u2264, >) for the null and alternative hypotheses.\r\nH0<\/sub><\/em>: \u03bc<\/em> __ 45 Ha<\/sub><\/em>:\u03bc<\/em> __ 45\r\n\r\n[reveal-answer q=\"889041\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"889041\"]\r\n\r\nH0<\/sub><\/em>: \u03bc<\/em> \u2265 45\r\n\r\nHa<\/sub><\/em>: \u03bc<\/em> < 45\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    <\/h3>\r\n
    \r\n

    Example<\/h3>\r\nIn an issue of U.S. News and World Report<\/em>, an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.\r\n\r\nH0<\/sub><\/em>: p<\/em> \u2264 0.066\r\n\r\nHa<\/sub><\/em>: p<\/em> > 0.066\r\n\r\n<\/div>\r\n

    <\/h3>\r\n
    \r\n

    try it<\/h3>\r\nOn a state driver's test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, \u2260, \u2265, <, \u2264, >) for the null and alternative hypotheses.\r\nH0<\/sub><\/em>: p<\/em> __ 0.40 Ha<\/sub><\/em>: p<\/em> __ 0.40\r\n\r\n[reveal-answer q=\"273064\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"273064\"]\r\n\r\nH0<\/sub><\/em>: p<\/em> = 0.40\r\n\r\nHa<\/sub><\/em>: p<\/em> > 0.40\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n
    \r\n\r\nWhen you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis\u00a0H0<\/sub><\/em> and the decision to reject or not. The outcomes are summarized in the following table:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    ACTION<\/th>\r\nH0<\/sub><\/em> IS ACTUALLY<\/th>\r\n...<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    <\/td>\r\nTrue<\/td>\r\nFalse<\/td>\r\n<\/tr>\r\n
    Do not reject\r\nH0<\/sub><\/em><\/td>\r\nCorrect Outcome<\/td>\r\nType II error<\/td>\r\n<\/tr>\r\n
    Reject\r\nH0<\/sub><\/em><\/td>\r\nType I Error<\/td>\r\nCorrect Outcome<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe four possible outcomes in the table are: The decision is\u00a0not to reject <\/strong>H<\/strong>0<\/strong><\/sub><\/em> when H0<\/sub><\/em> is true (correct decision)<\/strong>. The decision is to reject <\/strong>H<\/strong>0<\/strong><\/sub><\/em> when H0<\/sub><\/em> is true <\/strong>(incorrect decision known as a Type I error<\/strong>). The decision is not to reject <\/strong>H<\/strong>0<\/strong> <\/sub><\/em>when, in fact, H0<\/sub><\/em> is false<\/strong> (incorrect decision known as a Type II error<\/strong>). The decision is to reject <\/strong>H<\/strong>0<\/strong><\/sub><\/em> when H0<\/sub><\/em> is false<\/strong> (correct decision<\/strong> whose probability is called the Power of the Test<\/strong>).\r\n\r\nEach of the errors occurs with a particular probability. The Greek letters\r\n\u03b1<\/em> and \u03b2 <\/em>represent the probabilities.\r\n\r\n\u03b1<\/em> = probability of a Type I error = P<\/em>(Type I error)<\/strong> = probability of rejecting the null hypothesis when the null hypothesis is true.\r\n\r\n\u03b2<\/em> = probability of a Type II error = P<\/em>(Type II error)<\/strong> = probability of not rejecting the null hypothesis when the null hypothesis is false.\r\n\r\n\u03b1<\/em> and \u03b2<\/em> should be as small as possible because they are probabilities of errors. They are rarely zero.\r\n\r\nThe Power of the Test is 1 \u2013\u03b2<\/em>. Ideally, we want a high power that is as close to one as possible. Increasing the sample size can increase the Power of the Test.\r\n\r\nSuppose the null hypothesis,\u00a0H0<\/sub><\/em>, is: Frank's rock climbing equipment is safe.\r\n
      \r\n \t
    • Ty<\/strong>pe I error:<\/strong> Frank thinks that his rock climbing equipment may not be safe when, in fact, it really is safe.<\/li>\r\n \t
    • Type II error:<\/strong> Frank thinks that his rock climbing equipment may be safe when, in fact, it is not safe.<\/li>\r\n<\/ul>\r\n\u03b1<\/em> = probability<\/strong> that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is safe. \u03b2<\/em> = probability<\/strong> that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe.\r\n\r\nNotice that, in this case, the error with the greater consequence is the Type II error. (If Frank thinks his rock climbing equipment is safe, he will go ahead and use it.)\r\n
      \r\n

      try it<\/h3>\r\nSuppose the null hypothesis,\u00a0H0<\/em>, is: the blood cultures contain no traces of pathogen X<\/em>. State the Type I and Type II errors.\r\n\r\n[reveal-answer q=\"328698\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"328698\"]\r\n
        \r\n \t
      • Type I error: The researcher thinks the blood cultures do contain traces of pathogen X<\/em>, when in fact, they do not.<\/li>\r\n \t
      • Type II error: The researcher thinks the blood cultures do not contain traces of pathogen X<\/em>, when in fact, they do.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSuppose the null hypothesis,\u00a0H0<\/sub><\/em>, is: The victim of an automobile accident is alive when he arrives at the emergency room of a hospital.\r\n
          \r\n \t
        • Type I error:<\/strong> The emergency crew thinks that the victim is dead when, in fact, the victim is alive.<\/li>\r\n \t
        • Type II error:<\/strong> The emergency crew does not know if the victim is alive when, in fact, the victim is dead.<\/li>\r\n<\/ul>\r\n\u03b1<\/em> = probability<\/strong> that the emergency crew thinks the victim is dead when, in fact, he is really alive = P<\/em>(Type I error). \u03b2<\/em> = probability<\/strong> that the emergency crew does not know if the victim is alive when, in fact, the victim is dead =P<\/em>(Type II error).\r\n\r\nThe error with the greater consequence is the Type I error. (If the emergency crew thinks the victim is dead, they will not treat him.)\r\n\r\nhttps:\/\/youtu.be\/EowIec7Y8HM\r\n
          \r\n

          try it<\/h3>\r\nSuppose the null hypothesis,\u00a0H0<\/em>, is: a patient is not sick. Which type of error has the greater consequence, Type I or Type II?\r\n\r\n[reveal-answer q=\"354760\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"354760\"]\r\n\r\nThe error with the greater consequence is the Type II error: the patient will be thought well when, in fact, he is sick, so he will not get treatment.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIt is a Boy Genetic Labs claim to be able to increase the likelihood that a pregnancy will result in a boy being born. Statisticians want to test the claim. Suppose that the null hypothesis,\u00a0H0<\/em>, is: It's a Boy Genetic Labs has no effect on gender outcome.\r\n
            \r\n \t
          • Type I error: <\/strong>This results when a true null hypothesis is rejected. In the context of this scenario, we would state that we believe that It's a Boy Genetic Labs influences the gender outcome, when in fact it has no effect. The probability of this error occurring is denoted by the Greek letter alpha, \u03b1<\/em>.<\/li>\r\n \t
          • Type II error:<\/strong> This results when we fail to reject a false null hypothesis. In context, we would state that It's a Boy Genetic Labs does not influence the gender outcome of a pregnancy when, in fact, it does. The probability of this error occurring is denoted by the Greek letter beta, \u03b2<\/em>.<\/li>\r\n<\/ul>\r\nThe error of greater consequence would be the Type I error since couples would use the It's a Boy Genetic Labs product in hopes of increasing the chances of having a boy.\r\n
            \r\n

            try it<\/h3>\r\n\"Red tide\" is a bloom of poison-producing algae\u2013a few different species of a class of plankton called dinoflagellates. When the weather and water conditions cause these blooms, shellfish such as clams living in the area develop dangerous levels of a paralysis-inducing toxin. In Massachusetts, the Division of Marine Fisheries (DMF) monitors levels of the toxin in shellfish by regular sampling of shellfish along the coastline. If the mean level of toxin in clams exceeds 800 \u03bcg (micrograms) of toxin per kg of clam meat in any area, clam harvesting is banned there until the bloom is over and levels of toxin in clams subside. Describe both a Type I and a Type II error in this context, and state which error has the greater consequence.\r\n\r\n[reveal-answer q=\"765228\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"765228\"]\r\n\r\nIn this scenario, an appropriate null hypothesis would be\u00a0H0<\/sub><\/em>: the mean level of toxins is at most 800 \u03bc<\/em>g, H0<\/sub><\/em> : \u03bc<\/em>0<\/sub> \u2264 800 \u03bc<\/em>g.\r\n
              \r\n \t
            • Type I er<\/strong>ror: <\/strong>The DMF believes that toxin levels are still too high when, in fact, toxin levels are at most 800 \u03bc<\/em>g. The DMF continues the harvesting ban.<\/li>\r\n \t
            • Type II error:<\/strong> The DMF believes that toxin levels are within acceptable levels (are at least 800 \u03bc<\/em>g) when, in fact, toxin levels are still too high (more than 800 \u03bc<\/em>g). The DMF lifts the harvesting ban. This error could be the most serious. If the ban is lifted and clams are still toxic, consumers could possibly eat tainted food.<\/li>\r\n<\/ul>\r\nIn summary, the more dangerous error would be to commit a Type II error, because this error involves the availability of tainted clams for consumption.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nA certain experimental drug claims a cure rate of at least 75% for males with prostate cancer. Describe both the Type I and Type II errors in context. Which error is the more serious?\r\n
                \r\n \t
              • Type I:<\/strong> A cancer patient believes the cure rate for the drug is less than 75% when it actually is at least 75%.<\/li>\r\n \t
              • Type II:<\/strong> A cancer patient believes the experimental drug has at least a 75% cure rate when it has a cure rate that is less than 75%.<\/li>\r\n<\/ul>\r\nIn this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75% of the time, this most likely will influence the patient's (and doctor's) choice about whether to use the drug as a treatment option.\r\n
                \r\n

                try it<\/h3>\r\nDetermine both Type I and Type II errors for the following scenario:\r\n\r\nAssume a null hypothesis, H0<\/sub><\/em>, that states the percentage of adults with jobs is at least 88%.\r\n\r\nIdentify the Type I and Type II errors from these four statements.\r\n\r\na) Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%\r\n\r\nb) Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.\r\n\r\nc) Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.\r\n\r\nd) Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%.\r\n\r\n[reveal-answer q=\"727564\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"727564\"]\r\n\r\nType I error: c\r\n\r\nType I error: b\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n
                \r\n\r\nEarlier in the course, we discussed sampling distributions.\u00a0Particular distributions are associated with hypothesis testing.<\/strong> Perform tests of a population mean using a normal distribution<\/strong> or a Student's t-<\/em>distribution<\/strong>. (Remember, use a Student's t<\/em>-distribution when the population standard deviation<\/strong> is unknown and the distribution of the sample mean is approximately normal.) We perform tests of a population proportion using a normal distribution (usually n<\/em> is large or the sample size is large).\r\n\r\nIf you are testing a\u00a0single population mean<\/strong>, the distribution for the test is for means<\/strong>:\r\n\r\n[latex]\\displaystyle\\overline{{X}}\\text{~}{N}{\\left(\\mu_{{X}}\\text{ , }\\frac{{\\sigma_{{X}}}}{\\sqrt{{n}}}\\right)}{\\quad\\text{or}\\quad}{t}_{{{d}{f}}}[\/latex]\r\n\r\nThe population parameter is\u00a0\u03bc<\/em>. The estimated value (point estimate) for \u03bc is [latex]\\displaystyle\\overline{{x}}[\/latex], the sample mean.\r\n\r\nIf you are testing a\u00a0single population proportion<\/strong>, the distribution for the test is for proportions or percentages:\r\n\r\n[latex]\\displaystyle{P}^{\\prime}\\text{~}{N}{\\left({p}\\text{ , }\\sqrt{{\\frac{{{p}{q}}}{{n}}}}\\right)}[\/latex]\r\n\r\nThe population parameter is\u00a0p<\/em>. The estimated value (point estimate) for p<\/em> is p\u2032<\/em>. [latex]\\displaystyle{p}\\prime=\\frac{{x}}{{n}}[\/latex] where x<\/em> is the number of successes and n<\/em> is the sample size.\r\n

                Assumptions<\/h2>\r\nWhen you perform a\u00a0hypothesis test of a single population mean <\/strong>\u03bc<\/strong><\/em> using a Student's t<\/em>-distribution<\/strong> (often called a t-test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample<\/strong> that comes from a population that is approximately normally distributed<\/strong>. You use the sample standard deviation<\/strong> to approximate the population standard deviation. (Note that if the sample size is sufficiently large, a t-test will work even if the population is not approximately normally distributed).\r\n\r\nWhen you perform a\u00a0hypothesis test of a single population mean \u03bc<\/em> <\/strong>using a normal distribution (often called a z<\/em>-test), you take a simple random sample from the population. The population you are testing is normally distributed or your sample size is sufficiently large. You know the value of the population standard deviation which, in reality, is rarely known.\r\n

                When you perform a\u00a0hypothesis test of a single population proportion <\/strong>p<\/strong><\/em>, you take a simple random sample from the population. You must meet the conditions for a binomial distribution<\/strong> which are as follows: there are a certain number n<\/em> of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success p<\/em>. The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities np\u00a0<\/em>and nq<\/em> must both be greater than five (np<\/em> > 5 and nq<\/em> > 5). Then the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with \u03bc<\/em> = p<\/em> and [latex]\\displaystyle\\sigma=\\sqrt{{\\frac{{{p}{q}}}{{n}}}}[\/latex].<\/span> Remember that q<\/em> = 1 \u2013 p<\/em>.<\/p>\r\n\r\n

                Concept Review<\/h2>\r\nIn a\u00a0hypothesis test<\/strong>, sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis<\/strong>, typically denoted with H0<\/sub><\/em>. The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, \u2264 or \u2265) Always write the alternative hypothesis<\/strong>, typically denoted with Ha<\/sub><\/em> or H1<\/sub><\/em>, using less than, greater than, or not equals symbols, i.e., (\u2260, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.\r\n\r\nIn every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. A\u00a0Type I<\/strong> error occurs when a true null hypothesis is rejected. A Type II<\/strong> error occurs when a false null hypothesis is not rejected.\r\n\r\nThe probabilities of these errors are denoted by the Greek letters\u00a0\u03b1<\/em> and \u03b2<\/em>, for a Type I and a Type II error respectively. The power of the test, 1 \u2013 \u03b2<\/em>, quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable.\r\n\r\nIn order for a hypothesis test's results to be generalized to a population, certain requirements must be satisfied.\r\n\r\nWhen testing for a single population mean:\r\n
                  \r\n \t
                1. A Student's t<\/em>-test should be used if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation.<\/li>\r\n \t
                2. The normal test will work if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with a known standard deviation.<\/li>\r\n<\/ol>\r\nWhen testing a single population proportion use a normal test for a single population proportion if the data comes from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of success and the mean number of failures satisfy the conditions:\u00a0np<\/em> > 5 and nq<\/em> > n<\/em> where n<\/em> is the sample size, p<\/em> is the probability of a success, and q<\/em> is the probability of a failure.\r\n

                  Formula Review<\/h2>\r\nH0<\/sub><\/em> and Ha<\/sub><\/em> are contradictory.\r\n\r\n\u03b1<\/em> = probability of a Type I error = P<\/em>(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.\r\n\r\n\u03b2<\/em> = probability of a Type II error = P<\/em>(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.\r\n\r\nIf there is no given preconceived\u00a0\u03b1<\/em>, then use \u03b1<\/em> = 0.05.\r\n\r\nTypes of Hypothesis Tests<\/strong>\r\n
                    \r\n \t
                  • Single population mean, known<\/strong> population variance (or standard deviation): Normal test<\/strong>.<\/li>\r\n \t
                  • Single population mean, unknown<\/strong> population variance (or standard deviation): Student's t<\/em>-test<\/strong>.<\/li>\r\n \t
                  • Single population proportion: Normal test<\/strong>.<\/li>\r\n \t
                  • For a single population mean<\/strong>, we may use a normal distribution with the following mean and standard deviation. Means: [latex]\\displaystyle\\mu=\\mu_{{\\overline{{x}}}}{\\quad\\text{and}\\quad}\\sigma_{{\\overline{{x}}}}=\\frac{{\\sigma_{{x}}}}{\\sqrt{{n}}}[\/latex]<\/li>\r\n \t
                  • A single population proportion<\/strong>, we may use a normal distribution with the following mean and standard deviation. Proportions: [latex]\\displaystyle\\mu={p}{\\quad\\text{and}\\quad}\\sigma=\\sqrt{{\\frac{{{p}{q}}}{{n}}}}[\/latex].<\/li>\r\n<\/ul>","rendered":"
                    \n

                    Learning Outcomes<\/h3>\n
                    \n
                      \n
                    • Describe hypothesis testing in general and in practice<\/li>\n
                    • Differentiate between Type I and Type II Errors<\/li>\n
                    • Conduct and interpret hypothesis tests for a single population mean, population standard deviation known<\/li>\n
                    • Conduct and interpret hypothesis tests for a single population mean, population standard deviation unknown<\/li>\n<\/ul>\n<\/section>\n<\/div>\n

                      The actual test begins by considering two\u00a0hypotheses<\/strong>. They are called the null hypothesis<\/strong> and the alternative hypothesis<\/strong>. These hypotheses contain opposing viewpoints.<\/p>\n

                      H0<\/sub><\/em>: The null hypothesis:<\/strong> It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.<\/p>\n

                      Ha<\/sub><\/em>: The alternative hypothesis<\/strong>:<\/strong> It is a claim about the population that is contradictory to H0<\/sub><\/em> and what we conclude when we reject H0<\/sub><\/em>.<\/p>\n

                      Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.<\/p>\n

                      After you have determined which hypothesis the sample supports, you make a decision. There are two options for a\u00a0decision<\/strong>. They are “reject H0<\/sub><\/em>” if the sample information favors the alternative hypothesis or “do not reject H0<\/sub><\/em>” or “decline to reject H0<\/sub><\/em>” or “fail to reject H0<\/sub><\/em>” if the sample information is insufficient to reject the null hypothesis.<\/p>\n

                      Mathematical Symbols Used in\u00a0H0<\/sub><\/em> and Ha<\/sub><\/em>:<\/p>\n\n\n\n\n\n\n\n
                      H0<\/sub><\/em><\/th>\nHa<\/sub><\/em><\/th>\n<\/tr>\n<\/thead>\n
                      equal (=)<\/td>\nnot equal (\u2260)
                      \nor<\/strong> greater than (>) or<\/strong> less than (<)<\/td>\n<\/tr>\n
                      greater than or equal to (\u2265)<\/td>\nless than (<)<\/td>\n<\/tr>\n
                      less than or equal to (\u2264)<\/td>\nmore than (>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                      \n

                      Note<\/h4>\n

                      H0<\/sub><\/em> always has a symbol with an equal in it. Ha<\/sub><\/em> never has a symbol with an equal in it.<\/strong> The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.<\/p>\n