Problem 36 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose \(H_{0}: \quad p_{X}=p_{Y}\) is being tested against \(H_{1}: p_{X} \neq p_{Y}\) on the basis of two independent sets of one hundred Bernoulli trials. If \(x\), the number of successes in the first set, is sixty and \(y\), the number of successes in the second set, is forty-eight, what \(P\)-value would be associated with the data?

Step-by-Step Solution

Answer

The associated P-value with the data is approximately 0.0865.

Step 1: Calculate the Pooled Sample Proportion

First, to test the hypothesis, you must calculate the pooled sample proportion. The pooled sample proportion represents the success proportion in both samples combined. It is given by \((x + y) / (n_{1} + n_{2})\), where \(n_{1}\) and \(n_{2}\) represent the sample sizes. For this problem, \(n_{1} = n_{2} = 100\), \(x = 60\), and \(y = 48\). So, the pooled sample proportion \(\hat{p}\) is \((60 + 48) / (100 + 100) = 0.54\).

Step 2: Determine the Standard Error

Next, calculate the standard error (SE) of the difference in sample proportions. The standard error measures the statistical accuracy of an estimate. It's calculated as \(\sqrt{\hat{p} (1-\hat{p}) [(1/n_{1})+(1/n_{2})]}\). Substituting \(\hat{p} = 0.54\), \(n_{1} = 100\), and \(n_{2} = 100\), the SE is equal to \(\sqrt{0.54 * 0.46 * [ (1/100) + (1/100) ]} = 0.070\).

Step 3: Calculate the test statistic (z-score)

The test statistic or z-score measures how many standard deviations an element is from the mean. For two proportions, it's given by \((\hat{p}_{1} - \hat{p}_{2}) / SE\), where \(\hat{p_{1}} = x / n_{1}\) and \(\hat{p_{2}} = y / n_{2}\). Filling in the values for \(\hat{p_{1}} = 60 / 100 = 0.6\), \(\hat{p_{2}} = 48 / 100 = 0.48\), and \(SE = 0.070\), the test statistic is \(z = (0.6 - 0.48) / 0.070 = 1.714\).

Step 4: Find the P-value

Finally, you must calculate the \(P\)-value. Since it's a two-tailed test (\(p_{X} \neq p_{Y}\)), use the absolute value of \(z = 1.714\) and find the corresponding two-tailed \(P\)-value from the standard normal (Z) distribution table or using software. In this case, the \(P\)-value is approximately 0.0865.

Key Concepts

Bernoulli Trials

In statistics, Bernoulli trials refer to a sequence of experiments where each experiment, also known as trial, has exactly two possible outcomes, often termed success and failure. The probability of success is constant for each trial, and trials are independent of each other, meaning the outcome of one trial does not influence others.

For example, flipping a fair coin is a Bernoulli trial, where heads could be defined as success with a probability of 0.5, and tails as failure with the same probability. In hypothesis testing, like the exercise given, if each trial is checking for a specific attribute and classifying it as success or failure, then this series of trials constitutes Bernoulli trials.

Pooled Sample Proportion

When comparing proportions from two different samples, the pooled sample proportion is used. It is the weighted average of the sample proportions, assuming under the null hypothesis that the two populations have the same proportion (\( p_X = p_Y \) in this context).

Therefore, it is calculated by summing the number of successes in all groups and dividing by the total number of trials. The formula, \( (x + y) / (n_{1} + n_{2}) \), combines the sample data to provide an overall proportion. This pooled proportion serves as the estimated success rate for both groups which is particularly useful for hypothesis testing involving two sample proportions.

Standard Error

The standard error (SE) provides a measure of the variability in the sampling distribution of a statistic, in this case, the difference between two sample proportions. It reflects how much we would expect sample estimates to vary if we were to take multiple samples.

The smaller the standard error, the more certain we are about our sample estimate's representation of the true population value. In the exercise, the SE is computed using the pooled sample proportion with the formula \(\sqrt{\hat{p} (1-\hat{p}) [(1/n_{1})+(1/n_{2})]}\), indicating the uncertainty in the difference of the two sample proportions.

Test Statistic

The test statistic is a standardized value used to determine the extremeness of the observed result assuming the null hypothesis is true. In comparing two proportions, it's calculated by taking the difference between the two sample proportions and dividing by the standard error.

This gives us a z-score, which tells us how many standard deviations away from the expected proportion under the null hypothesis our observed difference is. The z-score in this problem assesses the magnitude of the difference between the sample proportions relative to the variability estimated by the standard error.

P-value

The P-value is a vital concept in hypothesis testing, representing the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, given that the null hypothesis is true.

It is not the probability that the null hypothesis is true, but it indicates how compatible our data is with the null hypothesis. A low P-value suggests that the null hypothesis may not adequately explain the observed data, prompting researchers to consider the alternative hypothesis. In our exercise, the P-value helps decide whether the observed difference between sample proportions is statistically significant.

Z-score

The z-score, also known as the standard score, quantifies the number of standard deviations an observation or statistic is from the mean. A z-score can reveal the direction and the magnitude of an outlier or how far an observation is from the hypothesized value.

Z-scores are a key part of finding probabilities in a standard normal distribution. They allow comparison between different sets of data, which may have different means and standard deviations. For hypothesis testing, the z-score transforms the test statistic to a standard form where it can be assessed against the standard normal distribution.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of zero and a standard deviation of one. All normal distributions can be transformed into a standard normal distribution using z-scores.

Test statistics calculated for hypothesis tests are often converted into z-scores so that they can be easily compared against the standard normal distribution to find areas under the curve, which translates into p-values. The areas under the curve represent probabilities, and thus, p-values for specific ranges of z-scores enable researchers to make inferences about their hypotheses.