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Chapter 20: Problem 1

A study of commuting times reports the travel times to work of a random sample of 1000 employed adults in Seattle. \({ }^{2}\) The mean is \(\mathrm{x}^{-} \bar{x}=\) 37.9 minutes and the standard deviation is \(s=27.2\) minutes. What is the standard error of the mean?

Short Answer

Expert verified
The standard error of the mean is approximately 0.86 minutes.

Step by step solution

01

Understand the Concept of Standard Error

The standard error of the mean (SEM) is a measure of how much the sample mean of the data is expected to vary from the true population mean. It provides an idea of the accuracy of the sample mean as an estimate of the population mean.
02

Identify the Formula for Standard Error of the Mean

The formula to calculate the standard error (SE) of the mean is given by \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the standard deviation of the sample and \( n \) is the sample size.
03

Plug Values into the Formula

In this problem, the standard deviation \( s = 27.2 \) and the sample size \( n = 1000 \). Plug these values into the formula: \( SE = \frac{27.2}{\sqrt{1000}} \).
04

Calculate the Square Root of the Sample Size

Compute the square root of the sample size \( n \): \( \sqrt{1000} \approx 31.62 \).
05

Compute the Standard Error

Divide the standard deviation \( s \) by the square root of the sample size: \( SE = \frac{27.2}{31.62} \approx 0.86 \). Thus, the standard error of the mean is approximately 0.86 minutes.

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Key Concepts

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

Standard Deviation
The standard deviation is a crucial concept in statistics when analyzing data. It measures the amount of variability or dispersion in a set of data values.
More simply, it tells us how much the values in our data set tend to deviate from the mean of the data set.
  • A small standard deviation indicates that the data points tend to be close to the mean.
  • A large standard deviation suggests that the data points are spread out over a wide range of values.
To calculate standard deviation, you first find the mean of the dataset. Then, subtract the mean from each data point to find the deviation of each value. Square each of these deviations, average them, and finally take the square root.
In the context of sampling, the standard deviation is not only descriptive but also instrumental in calculating the standard error, which helps estimate how well a sample mean represents the population mean.
Sample Size
Sample size refers to the number of observations or data points collected in a study. It plays a vital role in statistical analysis for a few reasons.
  • A larger sample size generally leads to more reliable and accurate estimates of population parameters.
  • It reduces the margin of error and increases the precision of estimates.
In our exercise, the sample size is 1000, which is quite substantial.
This large sample size is beneficial because it likely captures a diverse range of commuting times, leading to a better estimate of the actual average commuting time for all employed adults in Seattle.
Furthermore, the sample size directly impacts the standard error calculation: as the sample size increases, the denominator in the SEM formula grows, leading to a smaller standard error.
Population Mean Estimation
Population mean estimation is the process of using sample data to estimate the mean of an entire population. Since it is often impractical to measure an entire population, samples are used as proxies.
  • The sample mean is a point estimate of the population mean, giving a single value as the best estimate.
  • The standard error provides a measure of the sampling error involved in using the sample mean as an estimate for the population mean.
The smaller the standard error, the closer the sample mean is likely to be to the population mean.
In our case, the sample provides a mean of 37.9 minutes with a relatively small standard error of 0.86 minutes, indicating a reasonable estimate of the actual average commuting time.
Being vigilant about sample size and standard deviation, important components in the calculation, enhances the credibility of our population mean estimation.

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Most popular questions from this chapter

We prefer the \(t\) procedures to the z procedures for inference about a population mean because (a) \(z\) requires that you know the observations are from a Normal population, while \(t\) does not. (b) \(z\) requires that you know the population standard deviation \(\sigma\), while \(t\) does not. (c) \(z\) requires that you can regard your data as an SRS from the population, while \(t\) does not.

Cola makers test recipes for loss of sweetness during storage.Trained tasters rate the sweetness before and after storage. Here are the sweetness losses (sweetness before storage minus sweetness after storage) found by 10 tasters for a cola recipe currently on the market: \(\begin{array}{llllllllll}1.6 & 0.4 & 0.5 & -2.0 & 1.5 & -1.1 & 1.3 & -0.1 & -0.3 & 1.2\end{array}\) Take the data from these 10 carefully trained tasters as an SRS from a large population of all trained tasters. (a) Use these data to see if there is good evidence that the cola lost sweetness. (b) It is not uncommon to see the \(t\) procedures used for data like these. However, you should regard the results as only rough approximations. Why?

Does a football filled with helium travel farther than one filled with ordinary air? To test this, the Columbus Dispatch conducted a study. Two identical footballs, one filled with helium and one filled with ordinary air, were used. A casual observer was unable to detect a difference in the two footballs. A novice kicker was used to punt the footballs. A trial consisted of kicking both footballs in a random order. The kicker did not know which football (the helium-filled or the air-filled football) he was kicking. The distance of each punt was recorded. Then another trial was conducted. A total of 39 trials were run. Here are the data for the 39 trials, in yards that the footballs traveled. The difference (helium minus air) is the response variable. \({ }^{25}\) $$ \begin{array}{l|rrrrrrrrrr} \hline \text { Helium } & 25 & 16 & 25 & 14 & 23 & 29 & 25 & 26 & 22 & 26 \\ \text { Air } & 25 & 23 & 18 & 16 & 35 & 15 & 26 & 24 & 24 & 28 \\ \hline \text { Difference } & 0 & -7 & 7 & -2 & -12 & 14 & -1 & 2 & -2 & -2 \\\ \hline \text { Helium } & 12 & 28 & 28 & 31 & 22 & 29 & 23 & 26 & 35 & 24 \\ \text { Air } & 25 & 19 & 27 & 25 & 34 & 26 & 20 & 22 & 33 & 29 \\ \hline \text { Difference } & -13 & 9 & 1 & 6 & -12 & 3 & 3 & 4 & 2 & -5 \\ \hline \text { Helium } & 31 & 34 & 39 & 32 & 14 & 28 & 30 & 27 & 33 & 11 \\ \text { Air } & 31 & 27 & 22 & 29 & 28 & 29 & 22 & 31 & 25 & 20 \\ \hline \text { Difference } & 0 & 7 & 17 & 3 & -14 & -1 & 8 & -4 & 8 & -9 \\ \hline \text { Helium } & 26 & 32 & 30 & 29 & 30 & 29 & 29 & 30 & 26 & \\ \hline \text { Air } & 27 & 26 & 28 & 32 & 28 & 25 & 31 & 28 & 28 & \\ \hline \text { Difference } & -1 & 6 & 2 & -3 & 2 & 4 & -2 & 2 & -2 & \\ \hline \end{array} $$ (a) Examine the data. Is it reasonable to use the \(t\) procedures? (b) If your conclusion in part (a) is Yes, do the data give convincing evidence that the helium-filled football travels farther than the air-filled football?

In a randomized comparative experiment on the effect of color on the performance of a cognitive task, researchers randomly divided 69 subjects ( 27 males and 42 females ranging in age from 17-25 years) into three groups. Participants were asked to solve a series of six anagrams. One group was presented with the anagrams on a blue screen, one group saw them on a red screen, and one group had a neutral screen. The time, in seconds, taken to solve the anagrams was recorded. The paper reporting the study gives \(x-\bar{x}=11.58\) and \(s=4.37\) for the times of the 23 members of the neutral group. \({ }^{17}\) (a) Give a 95\% confidence interval for the mean time in the population from which the subjects were recruited. (b) What conditions for the population and the study design are required by the procedure you used in part (a)? Which of these conditions are important for the validity of the procedure in this case?

Which of these settings does not allow use of a matched pairs \(t\) procedure? (a) You interview both the instructor and one of the students in each of 20 introductory statistics classes and ask each how many hours per week homework assignments require. (b) You interview a sample of 15 instructors and another sample of 15 students and ask each how many hours per week homework assignments require. (c) You interview 40 students in the introductory statistics course at the beginning of the semester and again at the end of the semester and ask how many hours per week homework assignments require.
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