Another way to determine if a variable follows a normal distribution is to utilize the correlation statistic from a Normal Probability Plot (referred to as a QQ plot in StatCrunch). In this task, you will investigate whether the suicide rates (per 100,000) for the countries in the data table are normally distributed by using the QQ Plot, correlation statistic, and critical value.
Use StatCrunch to create a QQ plot for the variable: suicide rates (per 100,000). Be sure to include the correlation statistic.
What is the Critical Value for a sample of size 46? [Hint: Use the critical value for sample size n = 45.]
Answer:
Using the correlation statistic from the QQ Plot and the critical value from part b, is it reasonable to assume that the suicide rates (per 100,000) follow a normal distribution? Explain why or why not.
The Correct Answer and Explanation is :
To assess whether the suicide rates (per 100,000) across various countries follow a normal distribution, we can employ a Quantile-Quantile (QQ) plot using StatCrunch. This graphical tool helps visualize how the data compares to a normal distribution. Additionally, calculating the correlation coefficient from the QQ plot and comparing it to a critical value allows for a more formal assessment of normality.
a. Creating a QQ Plot in StatCrunch:
- Input Data: Enter the suicide rates into a single column in StatCrunch.
- Generate QQ Plot:
- Navigate to
Graph>QQ Plot. - Select the column containing the suicide rates.
- Check the option to display the correlation statistic.
- Click
Compute!to generate the plot.
The resulting QQ plot will display the observed data quantiles against the expected quantiles of a standard normal distribution. If the data is normally distributed, the points should align closely along a straight line.
b. Determining the Critical Value for a Sample Size of 46:
For a sample size of 46, we refer to the critical values of the normal probability plot correlation coefficient. According to the National Institute of Standards and Technology (NIST), the critical value for a sample size of 45 at the 5% significance level is approximately 0.973. (itl.nist.gov)
c. Assessing Normality:
After generating the QQ plot and obtaining the correlation coefficient (denoted as ‘r’), compare ‘r’ to the critical value:
- If ‘r’ > 0.973: The correlation exceeds the critical value, suggesting that it is reasonable to assume the data follows a normal distribution.
- If ‘r’ ≤ 0.973: The correlation does not exceed the critical value, indicating that the data may not be normally distributed.
Explanation:
A QQ plot is a valuable tool for assessing normality by plotting observed data against a theoretical normal distribution. If the data is normally distributed, the points on the QQ plot will align closely with the reference line. Deviations from this line suggest departures from normality, such as skewness or kurtosis.
The correlation coefficient from the QQ plot quantifies the strength of the linear relationship between the observed and expected quantiles. By comparing this coefficient to a critical value, we can make a formal decision about normality. In this case, with a sample size of 46, the critical value is approximately 0.973. If the calculated correlation coefficient exceeds this value, it provides evidence that the data is approximately normally distributed.
It’s important to note that while the QQ plot and correlation coefficient provide useful insights, they are part of a broader toolkit for assessing normality. Other methods, such as the Shapiro-Wilk test or examining skewness and kurtosis, can offer additional perspectives. However, for many practical purposes, especially with moderate sample sizes, the QQ plot combined with the correlation coefficient comparison is a robust approach to evaluating normality.
In summary, by generating a QQ plot in StatCrunch, calculating the correlation coefficient, and comparing it to the critical value of 0.973, we can assess whether it’s reasonable to assume that the suicide rates per 100,000 follow a normal distribution.