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Assume that the population of \(x\) values has an approximately normal distribution. Franchise: Candy Store Do you want to own your own candy store? With some interest in running your own business and a decent credit rating, you can probably get a bank loan on startup costs for franchises such as Candy Express, The Fudge Company, Karmel Corn, and Rocky Mountain Chocolate Factory. Startup costs (in thousands of dollars) for a random sample of candy stores are given below (Source: Entrepreneur Magazine, Vol. 23, No. 10 ). \(\begin{array}{lllllllll}95 & 173 & 129 & 95 & 75 & 94 & 116 & 100 & 85\end{array}\) (a) Use a calculator with mean and sample standard deviation keys to verify that \(\bar{x} \approx 106.9\) thousand dollars and \(s \approx 29.4\) thousand dollars. (b) Find a \(90 \%\) confidence interval for the population average startup costs \(\mu\) for candy store franchises. (c) Interpretation What does the confidence interval mean in the context of this problem?

Step by step solution

01

Calculate Mean and Sample Standard Deviation

To verify given values, first calculate the mean (\(\bar{x}\)) of the startup costs. Add all the sample values and divide by the number of samples (9): \(\bar{x} = \frac{95 + 173 + 129 + 95 + 75 + 94 + 116 + 100 + 85}{9} = 106.9\). Then, calculate the sample standard deviation (\(s\)), which measures the spread around the mean. Computing \(s\) manually or using a calculator confirms it is approximately 29.4.

02

Understand Confidence Interval Formula

A confidence interval estimates the range in which the population mean might lie. For a 90% confidence interval, the formula is: \(\bar{x} \pm z_{\alpha/2} \times \frac{s}{\sqrt{n}}\). Here, \(z_{\alpha/2}\) corresponds to the z-score for a 90% confidence level, \(s\) is the standard deviation, and \(n\) is the sample size.

03

Obtain the Z-Score for 90% Confidence

For a 90% confidence interval, \(z_{\alpha/2} = 1.645\). This is obtained from a standard normal distribution table or calculator.

04

Calculate the Confidence Interval

Plug values into the confidence interval formula: \(106.9 \pm 1.645 \times \frac{29.4}{3}\). Calculate the margin of error: \(1.645 \times 9.8 = 16.121\). Therefore, the confidence interval is \([106.9 - 16.121, 106.9 + 16.121]\) which is approximately \([90.779, 123.021]\).

05

Interpret the Confidence Interval

The 90% confidence interval suggests that if we repeated this sampling process many times, 90% of the calculated intervals would contain the true population mean startup cost. Here, we are 90% confident the true average startup cost is between $90,779 and $123,021.

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Key Concepts

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

Normal Distribution

The normal distribution is a key concept in statistics that refers to a bell-shaped, symmetrical curve. This distribution is important because many natural phenomena, including measurement errors and test scores, follow it. Its distinctive shape is defined such that most of the data in a set will fall near the mean (average), with fewer data points appearing as you move away from the center.
In the context of the candy store franchise example, we assume that the startup costs for candy stores are approximately normally distributed. This assumption allows us to apply statistical techniques, like confidence intervals, to infer about the population from a sample. This is crucial in practical scenarios where we cannot measure the whole population and need a reliable method to make predictions.

• The curve is symmetric around the mean.
• The mean, median, and mode of a normal distribution are equal.
• About 68% of values lie within one standard deviation of the mean.
• Approximately 95% are within two standard deviations, and 99.7% within three standard deviations.

Sample Standard Deviation

Sample standard deviation (\(s\)) is a measure that indicates how much individual data points in a sample deviate from the sample mean (\(\bar{x}\)). It's used to quantify the amount of variation or dispersion in a set of data points. When calculating it for a sample, rather than an entire population, it's known as the sample standard deviation.
This measure is particularly useful when making inferences or predictions based on just a part of the population.
In our case of startup costs for candy stores, the sample standard deviation is approximately 29.4 thousand dollars. This tells us about the spread or variability in the startup costs among the sampled stores. A higher value would indicate that costs are more spread out or varied, whereas a lower value suggests that they are more consistent.

Z-Score

A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is expressed as the number of standard deviations away from the mean a data point is. The formula to calculate a Z-score is:\[ Z = \frac{x - \bar{x}}{s} \]where \(x\) is the value being analyzed, \(\bar{x}\) is the mean, and \(s\) is the standard deviation.
In constructing confidence intervals, a specific Z-score is used to provide the margin of error for the estimate.
For our 90% confidence interval calculation, the Z-score is 1.645. This value is derived from the standard normal distribution and corresponds to a confidence level of 90%.
Essentially, this Z-score indicates how likely it is that the true mean falls within a certain range. Understanding Z-scores helps us estimate the probability of a statistic occurring within a normal distribution, which is a vital part of performing inferential statistics.

Population Mean

The population mean (\(\mu\)) is the average of a set of values for the whole population. It represents the central tendency of the overall data. In many cases, the population mean isn't directly measurable, especially in large populations.
Instead, we often use the sample mean (\(\bar{x}\)) to make inferences about the population mean. This is where confidence intervals come into play, using sample data to estimate the true population mean with a given level of certainty.

• In the candy store franchise example, we calculated a sample mean of \(106.9\) thousand dollars, which we use to estimate the population mean.
• Through our calculations, we determined that we are 90% confident that the actual average startup cost is between 90.779 and 123.021 thousand dollars.

Mathematically, confidence intervals help define how closely the sample mean approximates the population mean, providing an essential tool for making informed business and social science decisions.