{"id":337,"date":"2016-04-21T22:43:40","date_gmt":"2016-04-21T22:43:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstats1xmaster\/?post_type=chapter&p=337"},"modified":"2016-04-21T22:43:40","modified_gmt":"2016-04-21T22:43:40","slug":"distribution-needed-for-hypothesis-testing","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-fmcc-introstats1\/chapter\/distribution-needed-for-hypothesis-testing\/","title":{"raw":"Distribution Needed for Hypothesis Testing","rendered":"Distribution Needed for Hypothesis Testing"},"content":{"raw":"

Earlier 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).\n\nIf you are testing a\u00a0single population mean<\/strong>, the distribution for the test is for means<\/strong>:\n\n[latex]\\displaystyle\\overline{{X}}~{N}{\\left(\\mu_{{X}}\\frac{{\\sigma_{{X}}}}{\\sqrt{{n}}}\\right)}{\\quad\\text{or}\\quad}{t}_{{{d}{f}}}[\/latex]\n\nThe population parameter is\u00a0\u03bc<\/em>. The estimated value (point estimate) for \u03bc is [latex]\\displaystyle\\overline{{x}}[\/latex], the sample mean.\n\nIf you are testing a\u00a0single population proportion<\/strong>, the distribution for the test is for proportions or percentages:\n\n[latex]\\displaystyle{P}\\prime~{N}{\\left({p}\\sqrt{{\\frac{{{p}{q}}}{{n}}}}\\right)}[\/latex]\n\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.\n<\/p>

Assumptions<\/h2>\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).\n\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.\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>\n\n

Concept Review<\/h2>\nIn order for a hypothesis test's results to be generalized to a population, certain requirements must be satisfied.\n\nWhen testing for a single population mean:\n
  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>\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>\n<\/ol>\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.\n

    Formula Review<\/h2>\nIf there is no given preconceived\u00a0\u03b1<\/em>, then use \u03b1<\/em> = 0.05.\n\nTypes of Hypothesis Tests<\/strong>\n