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  • 1. COMPUTER-AIDED DATA ANALYSIS BOOK (978- 978-088-568-7) CHAPTER 11 One Sample t Test A one-sample t-test tests the difference between a sample mean and a known or hypothesized value (the test mean). The test uses the standard deviation of the sample to estimate σ (the population standard deviation). If the difference between the sample mean and the test mean is large relative to the variability of the sample mean, then µ is unlikely to be equal to the test mean. The One-Sample t test procedure, in essence, determines whether the mean of a single variable differs from a specified constant. It tests the hypothesis that a population has a particular mean value, e.g. zero. Ho: µ = 0 Sample mean is equal to a specified constant i.e. There is no significant difference (Difference not significant) Hi; µ ≠ 0 Sample mean is not equal to a specified constant (Two tailed) There is significant difference Hi; µ ≠ 0 is two directional, meaning that sample mean can be greater or less than a specified constant Hi: µ < > 0 (Two-tailed) Ho: µ ≤ ≥ 0 Sample mean is less than or equal to/ greater than or equal to a specified constant Hi: µ < > 0 Sample mean is less than / or greater than a specified constant (One tailed) A one-sample t test compares the mean of a single column of numbers against a hypothetical mean you entered. SPSS calculates the t ratio from this equation: = Sampling Error/Standard Error of the Mean Statistics - For each test variable: mean, standard deviation, and standard error of the mean. The average difference between each data value and the hypothesized test value, a t test that tests that this difference is 0, and a confidence interval for this difference (you can specify the confidence level A P value is computed from the t ratio and the numbers of degrees of freedom (which equals sample size minus 1). 11.1. Why use a one-sample t-test A one-sample t-test can help answer questions such as: 1
  • 2.  Does mean weight differ from the hypothetical test value (1.3 kg in this case)  Is the average IQ score different from a test value of say, 100?  Is the mean transaction time on target?  Does customer service meet expectations?  Does a water parameter meet WHO standard?  Is property rent inflation significant over a given number of years? 11.2. Checklist for using a one-sample t test Before accepting the results of any statistical test, first think carefully about whether you chose an appropriate test. Before accepting results from a one-sample t test, ask yourself these questions: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 11.3. SPSS Applications 11.3.1. To Obtain a One-Sample t Test: From the menus choose: Analyze > Compare Means > One-Sample t Test... Select one or more variables to be tested against the same hypothesized value. Enter a numeric test value against which each sample mean is compared. Optionally, you can click Options to control the treatment of missing data and the level of the confidence interval. 11.3.2. One-Sample t Test Data Considerations Data - To test the values of a quantitative variable against a hypothesized test value, choose a quantitative variable and enter a hypothesized test value. Assumptions. This test assumes that the data are normally distributed; however, this test is fairly robust to departures from normality. 11.3.3. One-Sample t Test Options >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 11.3.4. To Specify Options for One-Sample t Test From the menus choose: Analyze > Summarize > One-Sample t Test. In the One-Sample t Test dialog box, click Options. 2
  • 3. 11.4 SPSS DEMO. The data we will use is given in the table below, with the numbers indicating total protein (µg/ml). For our data, double click on the var at the top of the first column or click on the Variable View tab at the bottom of the page, type in “protein” in the Name column, and hit Enter. Under the assumption that you are going to enter numerical data, the rest of the row is filled in. Changes in the type and display of the variable can be made by clicking in the appropriate cells and using any buttons given. Then hit the Data View tab and type in the data values, following each by Enter. Save the file as usual where you wish under the name protein.sav. You just need type protein. The suffix is attached automatically. We wish to test whether the mean of the population from which the sample came is 70 as opposed to a true mean greater than 70. We test: Ho : µ = 70 Ha : µ > 70. From the menu, choose Analyze > Compare Means > One-Sample t Test. Select protein from the left-hand window and click the right arrow to move it to the Test Variable(s) window. Set the Test Value to 70. 3
  • 4. Click on Options. Set the Confidence Interval to 95% (or any other value you desire. Then click Continue followed by OK. You get the following output. 4
  • 5. SPSS gives us the basic descriptives in the first table. In the second table, we are given that the t-value for our test is 1.110. The p-value (or Sig. (2-tailed) is given as .272. Thus the p-value for our one-tailed test is one-half of that or .136. Based on this test statistic, we would not reject the null hypothesis, for instance, for a value of α = .05. SPSS also gives us the 95% Confidence Interval of the Difference between our data scores and the hypothesized mean of 70, namely (-2.6714, 9.3298). Adding the hypothesized value of 70 to both numbers gives us a 95% confidence interval for the mean of (67.3286, 79.3298). If you are only interested in the confidence interval from the beginning, you can just set the Test Value to 0 instead of 70. 11.5. Interpretations To make a decision, choose the significance level, α (alpha), before the test: • If P is less than or equal to α, reject Ho. • If P is greater than α, fail to reject Ho. (Technically, you never accept Ho, you simply fail to reject it.) A typical value for α is 0.05, but you can choose higher or lower values depending on the sensitivity required for the test and the consequences of incorrectly rejecting the null hypothesis. P (.136) > α (.05) we can accept the Ho and conclude that the difference is not significant i.e. the mean is equal to 70 P-value The t-test results indicate that >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 11.5.1 If the P value is small (one-sample t test) If the P value is small (usually defined to mean less than 0.05), then it is unlikely that the discrepancy you observed between sample mean and hypothetical mean is due to a coincidence arising from random sampling. You can reject the idea that the difference is a coincidence, and conclude instead that the population has a mean different than the hypothetical value you entered. The difference is statistically significant. But is the difference scientifically significant? The confidence interval helps you decide. 5
  • 6. The true difference between population mean and hypothetical mean is probably not the same as the difference observed in this experiment. There is no way to know the true difference between the population mean and the hypothetical mean. Prism presents the uncertainty as a 95% confidence interval. You can be 95% sure that this interval contains the true difference between the overall (population) mean and the hypothetical value you entered. 11.5.2 If the P value is large (one sample t test) If the P value is large, the data do not give you any reason to conclude that the overall mean differs from the hypothetical value you entered. This is not the same as saying that the true mean equals the hypothetical value. You just don't have evidence of a difference. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Further Reading: Bryman, A.S and Cramer D. (1997). Quantitaive data analysis with SPSS for Windows: A guide for social scientists. New York: Routeledge <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<  To purchase e-copy online, go to  Prospective purchasers in Nigeria should pay N1950= through GTB or UBA Accounts and provide evidence of payment.  Hard copies available soon 6
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