91Ó°ÊÓ

A research firm supplies manufacturers with estimates of the sales of their products from samples of stores. Marketing managers often look at the sales estimates and ignore sampling error. An SRS of 50 stores this month shows mean sales of 41 units of a particular appliance with standard deviation of 11 units. During the same month last year, an SRS of 52 stores gave mean sales of 38 units of the same appliance with a standard deviation of 13 units. An increase from 38 to 41 is a rise of \(7.9 \%\). The marketing manager is happy because sales are up \(7.9 \%\). (a) Give a 95\% confidence interval for the difference in mean number of units of the appliance sold at all retail stores. (b) Explain in language that the manager can understand why he cannot be confident that sales rose by \(7.9 \%\) and, in fact, may have dropped.

Step by step solution

01

Identify the Population Parameters

We are given two sample means and standard deviations from different years for sales data. Let's denote X as the sales from this year and Y as the sales from last year. We need to find a 95% confidence interval for the difference in means \( \mu_X - \mu_Y \).

02

Write Down the Sample Information

For this year: Sample size (n1) = 50, Mean (\( \bar{X} \)) = 41, Standard deviation (s1) = 11. Last year: Sample size (n2) = 52, Mean (\( \bar{Y} \)) = 38, Standard deviation (s2) = 13.

03

Determine the Formula for the Confidence Interval

Use the formula for the confidence interval for the difference in means: \[ (\bar{X} - \bar{Y}) \pm z^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \] where \( z^* \) is the z-value corresponding to the 95% confidence level.

04

Find the Z-value

For a 95% confidence interval, the critical z-value \( z^* \) is approximately 1.96.

05

Calculate the Confidence Interval

Substitute the known values into the formula: \[ (41 - 38) \pm 1.96 \sqrt{\frac{11^2}{50} + \frac{13^2}{52}} \] \[ = 3 \pm 1.96 \sqrt{\frac{121}{50} + \frac{169}{52}} \] \[ = 3 \pm 1.96 \times 2.237 \] \[ \approx 3 \pm 4.38 \] Thus, the interval is \[ -1.38 \text{ to } 7.38 \].

06

Interpret the Confidence Interval

The 95% confidence interval for the true difference in means is \(-1.38\) to \(7.38\). This implies that we are 95% confident that the true difference in sales means could be as low as \(-1.38\) units or as high as \(7.38\) units.

07

Explain to the Marketing Manager

The marketing manager should understand that the sales increase of \(7.9\%\) is not certain due to sampling variability. The confidence interval indicates that it is plausible the sales did not increase and might have even decreased, as evidenced by the interval including negative values.

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Key Concepts

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

Sampling Error

When you take a sample from a larger population, you often encounter what we call a "sampling error." This is the discrepancy between a sample statistic, like a mean or percentage, and the actual population parameter. Because you only have a portion of the data, your sample might not perfectly reflect the entire population.
For instance, if we consider the exercise about the sales of appliances, we are looking at sample data from some stores rather than every single store. Hence, there is a level of uncertainty about how well the sample mean represents the true average sales.
In essence, sampling error is the natural variation you expect due to only having a subset of the total population data. It cannot be completely avoided, but measures like larger sample sizes can help reduce it.

Difference in Means

The difference in means is a measure used when comparing two separate groups, to see how much their averages differ. In our appliance sales example, we have two samples: sales from this year and last year. The means of these samples are 41 and 38, respectively.
To find the difference in means, you subtract the mean of last year's sample from this year's sample mean, giving you 3 units. However, this result alone doesn’t confirm an increase in sales because of the inherent sampling error and variability.

• Mean of this year’s sample: 41 units
• Mean of last year’s sample: 38 units
• Difference in means: 3 units

To make informed conclusions, we calculate confidence intervals to understand the range in which the true difference might lie.

Z-value

When constructing a confidence interval, the z-value plays a crucial role. It is a number from the standard normal distribution that helps us determine how many standard deviations above or below the mean a data point lies. For a typical 95% confidence interval, the z-value is approximately 1.96.
This value allows us to gauge the reasonably expected variation in data under the assumption of a normal distribution.
In the example exercise, the z-value helps define the range around the sample mean difference where the true population mean difference could fall. Think of the z-value as a tool that helps translate statistical variability into the boundaries that form a confidence interval.

Critical Value

The critical value is a vital part of hypothesis testing and confidence interval calculation. It is the threshold that the test statistic must exceed to reject the null hypothesis, or the basis to construct a confidence interval.
In the sales example, for a 95% confidence interval, we used a critical z-value of 1.96. This critical value sets the width of our confidence interval, determining how much uncertainty we factor into our estimate of the difference in means.

• It helps to identify the range in which the true population parameter is likely to exist.
• A larger critical value broadens the interval, reflecting greater uncertainty.
• The choice of critical value often depends on the desired confidence level, like 95% or 99%.

By understanding the critical value, marketers and statisticians can better assess the reliability of their conclusions about changes in data like sales figures.