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Chapter 8: Problem 5

A random sample of the amount paid (in dollars) for taxi fare from downtown to the airport was obtained: $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline 15 & 19 & 17 & 23 & 21 & 17 & 16 & 18 & 12 & 18 & 20 & 22 & 15 & 18 & 20 \\ \hline \end{array}$$ Use the data to find a point estimate for each of the following parameters. a. Mean b. Variance c. Standard deviation

Short Answer

Expert verified
The point estimates for the given set of taxi fares are: \(\mu\) for the mean, \(\sigma^2\) for the variance and \(\sigma\) for the standard deviation. Calculating these using the formulas provided will provide the values.

Step by step solution

01

Calculate the Mean

Total the data set and divide by the number of data points to calculate the mean. There are 15 data points. The mean \(\mu\) is calculated as \(\mu = \frac{\text{{sum of all data points}}}{\text{{number of data points}}} = \frac{15+19+17+23+21+17+16+18+12+18+20+22+15+18+20}{15}\).
02

Calculate the Variance

Subtract the mean from each data point, square the result, then find the mean of these squared differences. The population variance \(\sigma^2\) is calculated as \(\sigma^2 = \frac{\text{{sum of the squared differences}}}{\text{{number of data points}}} = \frac{(15-\mu)^2 + (19-\mu)^2 + \cdots + (20-\mu)^2}{15}\).
03

Calculate the Standard Deviation

The population standard deviation \(\sigma\) is the square root of the variance. Therefore, \(\sigma = \sqrt{\sigma^2}\).

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Key Concepts

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

Mean Calculation
The mean, often referred to as the average, is a measure that represents the central value of a dataset. It is calculated by summing up all the observations and then dividing this total by the number of observations. In the context of the discussed exercise, the mean taxi fare from downtown to the airport is calculated by adding all the fares and then dividing by the number of fares, which is fifteen in this case.

For example, with our dataset, we add all the taxi fares together to get a sum. Then, this sum is divided by 15, giving us the mean taxi fare. The mean is useful as a point estimate because it gives us a single value that represents the entire dataset. However, it is sensitive to extreme values, often called 'outliers', which can skew the mean toward the higher or lower end of the scale.
Variance Calculation
Variance is a statistical measurement of the spread between numbers in a data set. More precisely, it measures how far each number in the set is from the mean and therefore from every other number in the set. To calculate the variance, we follow a two-step process.

First, for each number in the set, we subtract the mean and square the result. This step is crucial because squaring eliminates negative values and emphasizes larger differences. Second, we calculate the mean of these squared differences. In formula terms, the variance \( \sigma^2 \) for our taxi fare data would be the sum of these squared differences from the mean, divided by the number of data points.

Variance is valuable in understanding data dispersion, but because it is in squared units, it can be difficult to interpret in the context of the original data.
Standard Deviation Calculation
The standard deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance. It is denoted by \( \sigma \) for a population and \( s \) for a sample. In practice, the standard deviation provides insight into the concentration of data points around the mean.

The procedure to calculate the standard deviation starts with the calculation of the variance. Once the variance is determined, the standard deviation is obtained by taking the square root of the variance. This step essentially reverses the previous squaring of the deviations and brings the unit of measurement back to the original unit of the data points, making the standard deviation a very intuitive measure of spread which is easier to understand and apply than the variance.

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Most popular questions from this chapter

a. What decision is reached when the test statistic falls in the critical region? b. What decision is reached when the test statis- tic falls in the noncritical region?

When a parachute is inspected, the inspector is looking for anything that might indicate the parachute might not open. a. State the null and alternative hypotheses. b. Describe the four possible outcomes that can result depending on the truth of the null hypothesis and the decision reached. c. Describe the seriousness of the two possible errors.

A manufacturing process produces ball bearings with diameters having a normal distribution and a standard deviation of \(\sigma=0.04 \mathrm{cm} .\) Ball bearings that have diameters that are too small or too large are undesirable. To test the null hypothesis that \(\mu=0.50 \mathrm{cm},\) a sample of 25 is randomly selected and the sample mean is found to be 0.51. a. Design null and alternative hypotheses such that rejection of the null hypothesis will imply that the ball bearings are undesirable. b. Using the decision rule established in part a, what is the \(p\) -value for the sample results? c. If the decision rule in part a is used with \(\alpha=0.02\) what is the critical value for the test statistic?

Determine the value of the confidence coefficient \(z(\alpha / 2)\) for each situation described: a. \(1-\alpha=0.90\) b. \(1-\alpha=0.95\)

We are interested in estimating the mean life of a new product. How large a sample do we need to take to estimate the mean to within \(\frac{1}{10}\) of a standard deviation with \(90 \%\) confidence?
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