The file BabyWeight provides newborn weights for a simple random sample of 135 infants born in the United States in 1995. Can we conclude that the population from which this sample came is normally distributed?
a. Find the 10th, 20th, 30th, c percentiles of the standard normal distribution. For example, the 10th percentile of the standard normal distribution is –1.28155.
b. Standardize the sample data (subtract the sample mean and divide by the sample standard deviation). Place the sample data into categories based on the standard normal percentile groups, and report the number of data values in each category. For example, in the first group you would count the number of standardized sample observations that are less than or equal to –1.28155 to get the observed number of data points for that group. How many data points would you expect in each group?
c. Conduct a chi-square goodness-of-fit test and clearly state your conclusions. With this goodness-of-fit test, we are testing the form of the distribution instead of particular parameter values. Before doing many types of hypothesis tests or calculating confidence intervals, it is appropriate to determine if the data fit certain model assumptions, such as a certain distributional form. Often this process includes determining if the data were sampled from a normal distribution, as in this question. The steps in conducting a goodness-of-fit tests are as follows:
• Group the observed responses (y) into k arbitrarily chosen classes. In this exercise, we used evenly spaced percentiles of the normal distribution to create classes, but other classes (and more or fewer classes) could also be used. Classes must be distinct so that each observation can fall into only one class. If the result of the test is very sensitive to the choice of classes, then you cannot have much confidence in your conclusions.
• Calculate the expected frequencies based on some theoretical assumption about the distribution of the population (the random variable Y).
• Calculate the chi-square statistic. If the observed frequencies are very different from the expected frequencies, the test statistic will be large and we can reject the theoretical model. Note that we can never prove that the population actually follows a normal (or any other theoretical) model. We can only prove that a proposed model is wrong or fail to reject the model. Care should be taken not to state that we have proven that any population actually follows a particular distribution.