{"id":300,"date":"2021-07-14T15:59:09","date_gmt":"2021-07-14T15:59:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/test-of-a-single-variance\/"},"modified":"2023-12-05T09:43:40","modified_gmt":"2023-12-05T09:43:40","slug":"test-of-a-single-variance","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/test-of-a-single-variance\/","title":{"raw":"Hypothesis Test for Variance","rendered":"Hypothesis Test for Variance"},"content":{"raw":"
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Learning Outcomes<\/h3>\r\n
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  • Conduct a hypothesis test on one variance and interpret the conclusion in context<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n
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    Recall:\u00a0STANDARD DEVIATION AND VARIANCE<\/h3>\r\nThe most common measure of variation, or spread, is the standard deviation. The standard deviation is a number that measures how far data values are from their mean.\r\n\r\nTo calculate the standard deviation, we need to calculate the variance first. The variance is the average of the squares of the deviations [latex](x- \\overline{x})[\/latex] values for a sample, or the [latex]x \u2013 \u03bc[\/latex] values for a population). The symbol [latex]\\sigma ^2[\/latex] represents the population variance; the population standard deviation [latex]\u03c3[\/latex] is the square root of the population variance. The symbol [latex]s^2[\/latex] represents the sample variance; the sample standard deviation [latex]s[\/latex] is the square root of the sample variance.\r\n\r\nThe variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data.\r\n\r\n<\/div>\r\nA test of a single variance<\/strong> assumes that the underlying distribution is normal<\/strong>. The null and alternative hypotheses are stated in terms of the population variance<\/strong> (or population standard deviation). The test statistic is:\r\n

    [latex]\\displaystyle\\dfrac{\\left(n-1\\right)s^2}{\\sigma^2}[\/latex]<\/p>\r\n

    where:<\/p>\r\n\r\n

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    • [latex]n[\/latex]\u00a0= the total number of data<\/li>\r\n \t
    • [latex]s^2[\/latex]\u00a0= sample variance<\/li>\r\n \t
    • [latex]\\sigma^2[\/latex] = population variance<\/li>\r\n<\/ul>\r\nYou may think of [latex]s[\/latex]\u00a0as the random variable in this test. The number of degrees of freedom is [latex]df=n-1[\/latex]. A test of a single variance may be right-tailed, left-tailed, or two-tailed.<\/strong>\u00a0The example below\u00a0will show you how to set up the null and alternative hypotheses. The null and alternative hypotheses contain statements about the population variance.\r\n
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      Example 1<\/h3>\r\n
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      \r\n\r\nMath instructors are not only interested in how their students do on exams, on average, but how the exam scores vary. To many instructors, the variance (or standard deviation) may be more important than the average.\r\n\r\nSuppose a math instructor believes that the standard deviation for his final exam is five points. One of his best students thinks otherwise. The student claims that the standard deviation is more than five points. If the student were to conduct a hypothesis test, what would the null and alternative hypotheses be?\r\n[reveal-answer q=\"531530\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"531530\"]\r\n\r\nEven though we are given the population standard deviation, we can set up the test using the population variance as follows.\r\n\r\nH0<\/sub><\/em>: \u03c32<\/sup> = 52<\/sup>\r\nHa<\/sub><\/em>: \u03c32<\/sup> > 52<\/sup>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n
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      try it 1<\/h3>\r\n
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      \r\n\r\nA scuba instructor wants to record the collective depths of each of his students' dives during their checkout. He is interested in how the depths vary, even though everyone should have been at the same depth. He believes the standard deviation is three feet. His assistant thinks the standard deviation is less than three feet. If the instructor were to conduct a test, what would the null and alternative hypotheses be?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n
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      Recall: ORDER OF OPERATIONS<\/h3>\r\n
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      Please<\/strong><\/td>\r\nExcuse<\/strong><\/td>\r\nMy<\/strong><\/td>\r\nDear<\/strong><\/td>\r\nAunt<\/strong><\/td>\r\nSally<\/strong><\/td>\r\n<\/tr>\r\n
      parentheses<\/td>\r\nexponents<\/td>\r\nmultiplication<\/td>\r\ndivision<\/td>\r\naddition<\/td>\r\nsubtraction<\/td>\r\n<\/tr>\r\n
      [latex]( \\ )[\/latex]<\/td>\r\n[latex]x^2[\/latex]<\/td>\r\n[latex]\\times \\ \\mathrm{or} \\ \\div[\/latex]<\/td>\r\n[latex]+ \\ \\mathrm{or} \\ -[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

      To calculate the test statistic follow the following steps:<\/h2>\r\n1st find the numerator:<\/strong>\r\n\r\nStep 1: Calculate [latex](n-1)[\/latex] by reading the problem or counting the total number of data points and then subtract [latex]1[\/latex].\r\n\r\nStep 2: Calculate [latex]s^2[\/latex], and find the variance from the sample. This can be given to you in the problem or can be calculated with the following formula described in Module 2.\r\n\r\n[latex]s^2= \\frac{\\sum (x- \\overline{x})^2}{n-1}[\/latex]. Note if you are performing a test of a single standard deviation,\r\n\r\nStep 3: Multiply the values you got in Step 1 and Step 2.\r\n\r\nNote:<\/strong> if you are performing a test of a single standard deviation, in step 2, calculate the standard deviation, [latex]s[\/latex], by taking the square root of the variance.\r\n\r\n2nd find the denominator: <\/strong>If you are performing a test of a single variance, read the problem or calculate the population variance with the data. If you are performing a test of a single standard deviation, read the problem or calculate the population standard deviation with the data.\r\n\r\nFormula for the Population Variance: [latex]\\sigma ^2 = \\frac{\\sum (x- \\mu)^2}{N}[\/latex]\r\n\r\nFormula for the Population Standard Deviation: [latex]\\sigma = \\sqrt{\\frac{\\sum (x- \\mu)^2}{N}}[\/latex]\r\n\r\n3rd take the numerator and divide by the denominator.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n
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      EXAMPLE 2<\/h3>\r\n

      With individual lines at its various windows, a post office finds that the standard deviation for normally distributed waiting times for customers on Friday afternoon is 7.2 minutes. The post office experiments with a single, main waiting line and finds that for a random sample of 25 customers, the waiting times for customers have a standard deviation of 3.5 minutes.<\/p>\r\n

      With a significance level of 5%, test the claim that a single line causes lower variation among waiting times (shorter waiting times) for customers<\/strong>.\r\n[reveal-answer q=\"967369\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"967369\"]<\/p>\r\nSince the claim is that a single line causes less variation, this is a test of a single variance. The parameter is the population variance, \u03c3<\/em>2<\/sup>, or the population standard deviation, \u03c3<\/em>.\r\n\r\nRandom Variable:<\/strong> The sample standard deviation, s<\/em>, is the random variable. Let s<\/em> = standard deviation for the waiting times.\r\n\r\nH0<\/sub><\/em>: \u03c3<\/em>2<\/sup> = 7.22\r\nHa<\/sub><\/em>: \u03c3<\/em>2<\/sup> < 7.22\r\n\r\nThe word \"less\"<\/strong> tells you this is a left-tailed test.\r\n\r\nDistribution for the test:<\/strong>\u00a0[latex]X^2_{24}[\/latex], where:\r\n

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      • n<\/em> = the number of customers sampled<\/li>\r\n \t
      • df = n \u2013 1 = 25 \u2013 1 = 24<\/li>\r\n<\/ul>\r\nCalculate the test statistic:<\/strong>\r\n\r\nX<\/em>2<\/sup> = [latex]\\dfrac{\\left ( n-1 \\right )s^2}{\\sigma^2}[\/latex] = [latex]\\dfrac{\\left ( 25-1 \\right )\\left ( 3.5 \\right )^2}{7.2^2}[\/latex] = 5.67\r\n\r\nwhere n<\/em> = 25, s<\/em> = 3.5, and \u03c3<\/em> = 7.2.\r\n\r\nGraph:<\/strong>\r\n\r\n\"This\r\n\r\n \r\n\r\nProbability statement:<\/strong> p<\/em>-value = P<\/em> (\u03c72<\/sup> < 5.67) = 0.000042\r\n\r\nCompare \u03b1<\/em> and the p<\/em>-value:<\/strong>\r\n\r\n\u03b1<\/em> = 0.05; p<\/em>-value = 0.000042; \u03b1<\/em> > p<\/em>-value\r\n\r\nMake a decision:<\/strong> Since \u03b1<\/em> > p<\/em>-value, reject H0<\/sub><\/em>. This means that you reject \u03c3<\/em>2<\/sup> = 7.22. In other words, you do not think the variation in waiting times is 7.2 minutes; you think the variation in waiting times is less.\r\n\r\nConclusion:<\/strong> At a 5% level of significance, from the data, there is sufficient evidence to conclude that a single line causes a lower variation among the waiting times or with a single line, the customer waiting times vary less than 7.2 minutes.\r\n\r\nUsing a calculator:<\/strong>\r\n\r\nIn 2nd DISTR<\/code>, use 7:\u03c72cdf<\/code>. The syntax is (lower, upper, df)<\/code> for the parameter list. For this example, \u03c72<\/sup>cdf(-1E99,5.67,24)<\/code>. The p<\/em>-value = 0.000042.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n
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        try it 2<\/h3>\r\n
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        \r\n\r\nThe FCC conducts broadband speed tests to measure how much data per second passes between a consumer\u2019s computer and the internet. As of August 2012, the standard deviation of Internet speeds across Internet Service Providers (ISPs) was 12.2 percent. Suppose a sample of 15 ISPs is taken, and the standard deviation is 13.2. An analyst claims that the standard deviation of speeds is more than what was reported. State the null and alternative hypotheses, compute the degrees of freedom and the test statistic, sketch the graph of the p<\/em>-value, and draw a conclusion. Test at the 1% significance level.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n
        <\/section>","rendered":"
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        Learning Outcomes<\/h3>\n
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