91Ó°ÊÓ

The management of a supermarket chain wanted to investigate if the percentages of men and women who prefer to buy national brand products over the store brand products are different. A sample of 600 men shoppers at the company's supermarkets showed that 246 of them prefer to buy national brand products over the store brand products. Another sample of 700 women shoppers at the company's supermarkets showed that 266 of them prefer to buy national brand products over the store brand products. a. What is the point estimate of the difference between the two population proportions? b. Construct a \(95 \%\) confidence interval for the difference between the proportions of all men and all women shoppers at these supermarkets who prefer to buy national brand products over the store brand products. c. Testing at the \(5 \%\) significance level, can you conclude that the proportions of all men and all women shoppers at these supermarkets who prefer to buy national brand products over the store brand products are different?

Step by step solution

01

Calculation of Point Estimate

The point estimate of the difference between two population proportions is the difference between two sample proportions. For men, the proportion preferring national brand products is \(\frac{246}{600} = 0.41\). For women, it is \(\frac{266}{700} = 0.38\). Hence, point estimate is \(0.41 - 0.38 = 0.03\).

02

Construction of Confidence Interval

Let's denote the proportions and sample sizes as \(p1\), \(p2\), \(n1\), and \(n2\) respectively for better understanding. Here, \(p1 = 0.41\), \(n1 = 600\), \(p2 = 0.38\), and \(n2 = 700\). We will construct a \(95 \%\) confidence interval for the difference between the proportions.First, calculate the standard error \(SE\) using the formula:\[ SE = \sqrt{ \frac{p1(1-p1)}{n1} + \frac{p2(1-p2)}{n2} } \]Then calculate \(E\), the Margin of Error for a \(95\%\) confidence level is \(1.96 \times SE\). The Confidence interval is \((Point \ Estimate - E, Point \ Estimate + E)\).

03

Hypothesis Testing

Finally, we perform a hypothesis testing. For this, the null hypothesis \(H0\) is that the proportions are equal, i.e., \(p1=p2\), and the alternative hypothesis \(Ha\) is \(p1\neq p2\).The test statistic Z is calculated using the formula:\[ Z = \frac{(p1-p2)}{SE} \]Under the null hypothesis, Z follows a standard normal distribution. If the value of Z falls in the rejection region (i.e outside the range formed by -1.96 and +1.96 in this case), we reject the null hypothesis and conclude that the proportions of all men and all women shoppers who prefer to buy national brand products over the store brand products are different.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval

When conducting hypothesis tests, a confidence interval helps us determine the range within which we expect the true difference in population proportions to lie. It gives us an interval that is constructed around our point estimate. In this scenario, we want to see if the proportion difference of men and women who prefer national brands is significant.
To achieve this, we first need to calculate the standard error (SE) for our estimates. The smaller the SE, the more precise our estimates are. The margin of error (E) is then determined by multiplying the SE by the z-score corresponding to the desired confidence level, which is 1.96 for a 95% confidence interval. Adding and subtracting this margin of error from our point estimate results in the confidence interval.
This interval helps us understand the range of values we might expect if we repeatedly took samples from the same population.

Population Proportions

Population proportions refer to the ratio of individuals in our population with a particular attribute of interest. In this exercise, the focus is on understanding the proportion of men and women who prefer national over store brands.
For men, the observed proportion was calculated as 0.41, while for women, it was 0.38. By comparing these proportions, we can assess whether there's a significant difference in preferences.
The analysis involves using these sample proportions as estimates for their respective population proportions. Given that we work with samples rather than entire populations, it is crucial to be cautious about the reliability of these estimates, which leads us to further validation using standard error and hypothesis testing.

Standard Error

The standard error is a measure of the variability of our sample estimate, which reflects how much the sample proportions might fluctuate from the true population proportions. Calculating it allows us to understand the precision of our estimates of the population difference.
In this exercise scenario, the standard error is calculated using the formula:

• \[SE = \sqrt{ \frac{p1(1-p1)}{n1} + \frac{p2(1-p2)}{n2} }\]

Here, \(p1\) and \(p2\) are the sample proportions, and \(n1\) and \(n2\) are the sample sizes. This calculation provides us with a foundation to determine the confidence interval and conduct hypothesis testing. A smaller SE indicates that our estimate is more precise, which enhances the reliability of the inference we deduce from our data.

Significance Level

The significance level, often denoted as \(\alpha\), is a threshold we set to decide whether a statistical test result is significant. In this context, we use a 5% (0.05) significance level to test if the proportions of men and women preferring national brands differ.
This level defines our tolerance for incorrectly concluding that there is a difference when there isn't one, also known as a Type I error. By choosing a 5% significance level, we limit the likelihood of such errors to 5%.
Using this threshold, we calculate the test statistic Z from our sample proportions and standard error. We compare this Z value to determine if it falls in the "rejection region," typically bounded by critical values earmarked by the selected significance level. If it lies outside this range, we reject the null hypothesis and conclude that a significant difference exists between the proportions.