Problem 31 - An Introduction to Mathematical Statistics and Its Applications

Question

The phenomenon of handedness has been extensively studied in human populations. The percentages of adults who are right-handed, left-handed, and ambidextrous are well documented. What is not so well known is that a similar phenomenon is present in lower animals. Dogs, for example, can be either right-pawed or left-pawed. Suppose that in a random sample of two hundred beagles, it is found that fifty-five are left-pawed and that in a random sample of two hundred collies, forty are left-pawed. Can we conclude that the difference in the two sample proportions of left-pawed dogs is statistically significant for \(\alpha=0.05 ?\)

Step-by-Step Solution

Answer

Yes, the difference in the sample proportions of left-pawed beagles and collies is statistically significant at the \(\alpha = 0.05\) level.

Step 1: Calculate the Sample Proportions

Begin by calculating the sample proportions for beagles and collies which are left-pawed. For beagles, this is \(55/200 = 0.275\) and for collies, it is \(40/200 = 0.20\).

Step 2: Calculate the Pooled Sample Proportion

Next, calculate the pooled sample proportion of left-pawed dogs, which is the total number of left-pawed dogs divided by the total number of dogs. This will give us \((55 + 40)/(200 + 200) = 0.2375\).

Step 3: Calculate the Standard Error

The standard error (SE) for the difference of two proportions is calculated using the formula: \[SE = \sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}\] where \(n_1\) and \(n_2\) are the sample sizes. Substituting our known values, we find that \(SE = 0.034\).

Step 4: Calculate the Test Statistic

The test statistic, Z, is calculated by dividing the difference in sample proportions by the standard error. This gives us \(Z = (0.275 - 0.20)/0.034 = 2.21\).

Step 5: Compare Test Statistic to a Standard Normal Distribution

We need to compare our test statistic to a standard normal distribution with a level of significance of 0.05. This value in the standard normal distribution table is 1.96. Here our test statistic, 2.21, is more than 1.96. This indicates that the difference in proportions is statistically significant at the \(\alpha = 0.05\) level.

Key Concepts

Hypothesis Testing

Hypothesis Testing is a statistical method used to determine whether there is enough evidence to support a specific hypothesis about a population parameter. In our beagle and collie study, we're comparing whether the observed difference in the proportion of left-pawed dogs between two breeds is statistically significant or if it's just due to random chance.
To conduct a hypothesis test, you start by establishing two hypotheses:

• The null hypothesis (\( H_0 \)): Assumes no effect or difference. For our case, it would suggest that the proportion of left-pawed beagles is equal to the proportion of left-pawed collies.
• The alternative hypothesis (\( H_a \)): Indicates that there is an actual difference. Here, it would imply that proportions are not equal.

Once you compute the test statistic — in this case, using the difference in sample proportions and standard error — you compare it to a critical value from a statistical distribution. If your test statistic falls beyond this critical value, you reject the null hypothesis in favor of the alternative one.

Sample Proportion

A Sample Proportion is a statistic that estimates the proportion of a particular characteristic in a population from a subset known as a sample. In our example, we evaluated the sample proportions of left-pawed dogs in the beagle and collie populations.
To find the sample proportion, use the formula:\[ \hat{p} = \frac{x}{n}\]where:

• \(x\) = the number of success (e.g., left-pawed dogs)
• \(n\) = the total sample size

For beagles, the sample proportion is 0.275, indicating that 27.5% of sampled beagles are left-pawed. Similarly, for collies, it's 0.20, showing 20% are left-pawed. These sample proportions serve as the fundamental components of the hypothesis test.

Standard Error

The Standard Error is a measure of the statistical accuracy of an estimate, specifically, it quantifies how much sample proportions can be expected to vary from the true population proportion. It essentially tells us how much uncertainty we have around our estimate.
For the difference in two sample proportions, the standard error is calculated using the formula:\[SE = \sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}\]where:

• \(\hat{p}\) = pooled sample proportion
• \(n_1\) and \(n_2\) = sizes of the two samples

In our example of beagles and collies, the calculated standard error is 0.034, giving a sense of how much our observed proportions might naturally vary from sample to sample.

Pooled Sample Proportion

When comparing the proportions between two samples, a Pooled Sample Proportion provides a combined estimate of the population proportion, assuming no difference between the two samples. This can simplify calculations and improve estimate accuracy.
To find the pooled sample proportion, use:\[\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}\]where:

• \(x_1\) and \(x_2\) are the number of successes in each sample (e.g., left-pawed dogs).
• \(n_1\) and \(n_2\) are the total sample sizes of the groups.

For example, combining data from both beagles and collies yields a pooled proportion of 0.2375, indicating that 23.75% of the total dogs sampled are left-pawed. This pooled proportion is crucial for calculating the standard error used in the hypothesis test.