As in Example 1, we combine the first two and the last two intervals so that all frequencies are at least 5. We next test the null hypothesis that the data is normally distributed using the sample mean and variance (3.74 and 4.84 respectively as seen in Figure 3) as estimates for the population mean/variance. a weight gain of 1 kg) and 9 respectively.įigure 3 – Calculation of mean and standard deviation We first calculate the sample mean and variance as described in Frequency Tables using the midpoint of each interval, although for the first and last intervals (-∞,0] and [8,∞) we need to guess at acceptable representative values, which we take as -1 (i.e. This time we will simply ask whether the above data comes from a normal population. Note that the df = number of intervals – 1 = 8 – 1 = 7 since the mean and standard deviation are given.Įxample 2: In the above example, the population mean and variance were known. The chi-square test statistic is 4.47, which is less than the critical value of CHIINV(.05,7) = 14.07, and so we can conclude that there is a good fit. Since the observed and expected frequencies of the first and last interval are less than 5, it is better to combine the 1 st and 2 nd as well as the last and second to last intervals. We now perform the Chi-square goodness of fit test. Multiplying these figures by the sample size of 90, gives us the expected frequency.įigure 2 – Chi-square test with known mean and std deviation The probability that x is in the interval ( a, b] is then NORM.DIST( b, 4, 2.5, TRUE) – NORM.DIST( a, 4, 2.5, TRUE). This probability is NORM.DIST( b, 4, 2.5, TRUE). We begin by calculating the probability that x < b for b = 0, 1, …, 8, assuming a normal distribution with mean 4 and standard deviation 2.5. Test whether the data is normally distributed with mean 4 kg and standard deviation of 2.5 kg.įigure 1 – Frequency table and histogram for Example 1 The following frequency table shows the weight gain (in kilograms). H 0: data are sampled from a normal distribution.Įxample 1: 90 people were put on a weight gain program. In particular, we can use Theorem 2 of Goodness of Fit, to test the null hypothesis: The chi-square goodness of fit test can be used to test the hypothesis that data comes from a normal hypothesis.