Question

When Bargaining Pays Off According to a survey done by Consumer Reports, you should always try to negotiate for a better deal when shopping or paying for services. \(^{20}\) Tips include researching prices at other stores and on the Internet, timing your visit late in the month when salespeople are trying to meet quotas, and talking to a manager rather than a salesperson. Suppose that random samples of 200 men and 200 women are taken, and that the men were more likely than the women to say they "always or often" bargained (30\% compared with \(25 \%\) ). a. Construct a \(95 \%\) confidence interval for the differ- ence in the proportion of men and women who say they "always or often" negotiate for a better deal. b. Do the data indicate that there is a difference in the proportion of men and women who say they "always or often" negotiate for a better deal? Explain.

Short answer

Explain your reasoning. Answer: No, we cannot conclude that there is a significant difference between the proportions of men and women who say they "always or often" negotiate for a better deal. This is because the 95% confidence interval calculated (-0.0376, 0.1376) contains both positive and negative values, as well as 0. Since the interval contains 0, it indicates that the difference could be zero, meaning that there may be no actual difference between the proportions of men and women who bargain.

Step-by-step solution

Calculate the proportions

First, let's find the proportions of men and women who say they "always or often" bargain: Proportion of men (P1) = 0.30 Proportion of women (P2) = 0.25

Calculate the pooled proportion

Next, we'll find the pooled proportion (Pp), which is the proportion of both men and women combined who "always or often" bargain. Pooled proportion (Pp) = (Number of men who bargain + Number of women who bargain) / (Total number of men + Total number of women) Pp = (0.30 * 200 + 0.25 * 200) / (200 + 200) Pp = (60 + 50) / 400 Pp = 110 / 400 Pp = 0.275

Calculate the standard error

Now, we'll calculate the standard error (SE) of the difference in proportions: SE = sqrt[(Pp * (1 - Pp) / Number of men) + (Pp * (1 - Pp) / Number of women)] SE = sqrt[(0.275 * (1 - 0.275) / 200) + (0.275 * (1 - 0.275) / 200)] SE = sqrt[(0.199375 / 200) + (0.199375 / 200)] SE = sqrt[0.000996875 + 0.000996875] SE = sqrt[0.00199375] SE ≈ 0.0447

Find the critical value and calculate the confidence interval

For a 95% confidence interval, the critical value (Z) can be found using a Z-table, which is approximately 1.96. Now we can calculate the confidence interval as follows: Lower limit = (P1 - P2) - Z * SE Upper limit = (P1 - P2) + Z * SE Lower limit = (0.30 - 0.25) - 1.96 * 0.0447 Upper limit = (0.30 - 0.25) + 1.96 * 0.0447 Lower limit ≈ 0.05 - 0.0876 Upper limit ≈ 0.05 + 0.0876 Lower limit ≈ -0.0376 Upper limit ≈ 0.1376 So, the 95% confidence interval for the difference in proportions is approximately (-0.0376, 0.1376).

Analyze the confidence interval

The confidence interval contains both positive and negative values (-0.0376, 0.1376). This means that we are 95% confident that the difference in the proportion of men and women who say they "always or often" bargain could range from approximately 3.76% fewer men bargaining to 13.76% more men bargaining compared to women. Since the interval contains 0, we cannot conclude that there is a significant difference between the proportions of men and women who "always or often" negotiate for a better deal.

Key concepts

Proportion Difference

The term 'proportion difference' refers to the discrepancy between the proportions of two groups with respect to a certain characteristic or behavior. In studies comparing two groups, such as men and women who 'always or often' negotiate for better deals, understanding and calculating the proportion difference is crucial in determining if there is a variation in behavior between the two groups.

For example, consider the scenario involving the negotiation habits of men and women. To calculate the proportion difference, we subtract the proportion of women who negotiate (25%) from the proportion of men who negotiate (30%), leading to a 5% difference. This simple calculation highlights how often each group engages in bargaining, setting the stage for further statistical analysis. It's important to express these proportions accurately, as they directly impact the confidence intervals and hypothesis testing outcome which relate to assessing the statistical significance of the observed difference.

Statistical Significance

Statistical significance is a fundamental concept in analyzing data to understand if an observed difference or effect likethe proportion difference is likely to be due to chance or if it represents a true difference in the population. The significance is determined by calculating the probability of observing a result as extreme as the one measured, assuming that there were actually no difference (null hypothesis).

In the given negotiation example, once the proportion difference is determined, statisticians deploy a confidence interval to assess if the observed difference is statistically significant. The confidence interval is a range of values, derived from sample data, that could contain the true population difference. If a 95% confidence interval does not include 0 (indicating no difference), it suggests that the observed difference is statistically significant, and not a result of random chance. However, if the interval does include 0, as it does in the example (-0.0376 to 0.1376), it implies that the evidence is not strong enough to declare a statistically significant difference, and the result could be due to random variation.

Hypothesis Testing

Hypothesis testing is a systematic procedure used in statistics to evaluate two competing theories or hypotheses about a population, using sample data. The null hypothesis (\( H_0 \)) represents a general statement or default position that there is no relationship or difference between two measured phenomena. In contrast, the alternative hypothesis (\( H_1 \text{or}\ H_a \text{or}\ H_0 \text{and}\ H_1 \text{are a pair}\)) suggests that there is a relationship or difference.

In performing a hypothesis test, if the evidence within the sample data is strong enough to reject the null hypothesis in favor of the alternative hypothesis, we say that the result is statistically significant. This decision is often based on a pre-defined significance level (commonly 5%, or a p-value less than 0.05). In the exercise about bargaining behaviors, even with a calculated proportion difference, the hypothesis test revealed that the confidence interval included 0, which means we fail to reject the null hypothesis of no difference between the proportions of men and women who negotiate. Consequently, there isn't sufficient evidence to assert that one gender 'always or often' negotiates for a better deal more than the other.