Question

Let \(p\) denote the proportion of grocery store customers who use the store's club card. For a largesample \(z\) test of \(H_{0}: p=.5\) versus \(H_{a}: p>.5,\) find the \(P\) -value associated with each of the given values of the test statistic: a. 1.40 d. 2.45 b. 0.93 e. -0.17 c. 1.96

Short answer

The \(P\) -values for the given Z-scores are as follows: a) 0.0808, b) 0.1783, c) 0.0250, d) 0.0071, e) 0.5675.

Step-by-step solution

Understand the values and tables

Before deriving the \(P\) -value, the Z-score must be understood. The Z-score is a measure of how many standard deviations an element is from the mean. Therefore, the given values (1.40,0.93,1.96,2.45 and -0.17) are all Z-scores. To calculate the \(P\) -value, a Standard Normal Distribution Table, or Z-table, is needed. The Z-table provides the probability that a normally distributed random variable Z is less than or equal to a given value.

Finding the p-values (Unilateral test)

Because the alternative hypothesis \(H_{a}\) suggests that \(p>.5\), this is a one-sided test, and the \(P\) -values should be calculated as such. To find the \(P\) -value from the Z-table, one must locate the corresponding value and then subtract from 1, since we are looking for \(p>.5\). For instance, for Z=1.40, the Z-table gives a probability of 0.9192, so \(P=(1-0.9192)=0.0808\). The same strategy is applied to the remaining Z-scores.

Interpret the p-values

The \(P\) -value is the probability of observing a result as extreme as the test statistic, assuming the null hypothesis is true. If the \(P\) -value is less than the significance level, which is typically set at 0.05, then the null hypothesis is rejected. In this context, a lower \(P\) -value implies that more customers are likely to use the store's club card.

Key concepts

Z-test

The Z-test is a statistical method used to determine whether there is a significant difference between sample and population parameters. It is based on the assumption that the samples are drawn from a normal distribution. In the context of hypothesis testing, a Z-test may be used to ascertain whether an observed sample mean differs significantly from a known population mean, or in the case of proportions, whether the observed proportion differs from a hypothesized value.

The Z-test involves calculating a Z-score, which represents the number of standard deviations a data point is from the population mean. The Z-score is then used to find the P-value from the standard normal distribution to determine if the results are significant.

Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is the distribution that Z-scores are based on. In practical terms, the Z-score of any data point tells us how far and in what direction that data point is from the mean, measured in units of standard deviation. When all the Z-scores are plotted, they form the standard normal distribution. This curve is symmetrical, and most values are within three standard deviations from the mean.

Understanding the properties of the standard normal distribution is critical for performing Z-tests, as it serves as the reference for determining the probability (P-value) of observing a particular value of the test statistic under the null hypothesis.

Hypothesis Testing

Hypothesis testing is a method used in statistics to decide between two competing hypotheses about a population parameter, based on sampled data. The first step in hypothesis testing is to establish a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).

The null hypothesis is typically a statement of no effect or no difference, whereas the alternative hypothesis represents a new claim or the effect we wish to test. Using a sample, we calculate a test statistic (like the Z-score in Z-tests) and utilize this statistic to calculate the P-value, which helps us determine whether to reject the null hypothesis or not.

Null Hypothesis

The null hypothesis (\(H_0\)) is a statement used in statistical tests that assumes there is no significant difference or effect. It represents the 'status quo' and is the hypothesis that researchers seek to test against. In the exercise provided, the null hypothesis states that the proportion of customers who use the store's club card is 0.5, or in other words, half of the customers use the card. The null hypothesis serves as a baseline that the test statistic is compared against to determine the P-value.

Alternative Hypothesis

The alternative hypothesis (\(H_a\)) represents a statement that is accepted if the null hypothesis is rejected. It proposes what we suspect might be true instead of the null hypothesis. Unlike the null hypothesis, the alternative hypothesis indicates a new effect or change. In our example, the alternative hypothesis is that more than half (\(p > 0.5\)) of the grocery store customers use the store's club card. It provides direction for the statistical test and influences the type of test performed, such as one-tailed or two-tailed tests.

Significance Level

The significance level, often denoted by \(\alpha\), is a threshold used to determine whether or not to reject the null hypothesis. It is a probability value that defines the unlikely range for the P-value if the null hypothesis is true. Common significance levels are 0.01, 0.05, and 0.10. In most social science research, the significance level is set at 0.05, meaning there is a 5% risk of rejecting the null hypothesis when it is, in fact, true. This level is chosen to balance the risk of making an error with the need for reasonable evidence against the null hypothesis.

Z-table

A Z-table, also known as the standard normal table, is a mathematical table that allows us to find the probability that a standard normal variable falls within a specified range. The Z-table displays the cumulative probabilities of the standard normal distribution. As Z-scores follow the standard normal distribution, the table can be used in conducting Z-tests.

When you're doing a hypothesis test, you use the Z-table to find the P-value by locating the area to the left of the Z-score. If the test is one-tailed (as in our exercise, where we're interested in values greater than a certain point), you would subtract this area from 1 to get the P-value. For the given values in the exercise, you would find the corresponding cumulative probabilities for each Z-score and perform the appropriate calculations to determine the P-values.