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Chapter 9: Problem 85

The mean work week for engineers in a start-up company is believed to be about 60 hours. A newly hired engineer hopes that it's shorter. She asks ten engineering friends in start-ups for the lengths of their mean work weeks. Based on the results that follow, should she count on the mean work week to be shorter than 60 hours? Data (length of mean work week): 70; 45; 55; 60; 65; 55; 55; 60; 50; 55.

Short Answer

Expert verified
Yes, her sample suggests the mean work week is shorter than 60 hours.

Step by step solution

01

Organize the Data

Given the data of the lengths of the mean work weeks for 10 engineers: 70, 45, 55, 60, 65, 55, 55, 60, 50, 55. Arrange these values in a list for further calculations.
02

Calculate the Sample Mean

To find the average, sum all the given data points and divide by the number of data points. The calculation is as follows: \( \text{Mean} = \frac{70 + 45 + 55 + 60 + 65 + 55 + 55 + 60 + 50 + 55}{10} = \frac{570}{10} = 57 \).
03

Compare with the Hypothesized Mean

The hypothesized mean work week is 60 hours. Compare the calculated sample mean (57 hours) with this hypothesized mean. Since 57 < 60, the sample mean is indeed shorter than the believed mean.
04

Conclusion

Based on the sample mean calculated (57 hours), and the comparison made, it suggests that the mean work week among her friends is shorter than the believed average work week of 60 hours.

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Key Concepts

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

Sample Mean
When you want to find the average from a sample of data, the sample mean is your go-to tool. It's calculated by summing up all the data points in your set and then dividing by the number of data points you have. By doing this, you get an average that represents the central point of your data. In the exercise, the sample mean is calculated using the workweeks of ten engineers. The formula applied is:\[ \text{Sample Mean} = \frac{70 + 45 + 55 + 60 + 65 + 55 + 55 + 60 + 50 + 55}{10} = 57 \]This sample mean of 57 hours helps you to understand the typical workweek length among the sample of engineers surveyed.
Data Organization
Organizing your data effectively is an important first step in any analysis. It means taking raw data and putting it in order so you can understand it better and use it for calculations or comparisons later. For the exercise given, the workweek data were arranged in a list: 70, 45, 55, 60, 65, 55, 55, 60, 50, and 55.
  • Helps identify patterns in the data.
  • Makes calculations like the sample mean easier to carry out.
  • Clear organization aids in recognizing outliers or anomalies, if any.
Having your data organized ensures that your analysis can be done smoothly and accurately.
Comparison with Hypothesized Mean
Once you have calculated your sample mean, the next step is to compare it with a hypothesized value. In hypothesis testing, this is crucial because it helps determine whether there's a significant difference between your sample data and what you believed previously. In this specific exercise, the hypothesized mean is 60 hours. The sample mean we calculated was 57 hours. By comparing the two:
  • Sample Mean (57 hours) is less than Hypothesized Mean (60 hours).
  • This shows the average workweek among the sample is indeed shorter than anticipated.
This comparison allows the newly hired engineer to confirm or refute the initial assumption about work hours.
Conclusion Drawing
The final step in hypothesis testing is drawing a conclusion from the comparison results. Conclusions are essential because they guide decisions or offer insights based on the data analysis. Based on the sample mean of 57 hours, which is shorter than the hypothesized mean of 60 hours, one can conclude:
  • The mean work week among the engineer's friends is shorter.
  • It supports the engineer's hope of a shorter work week compared to the belief.
Conclusively, these findings enable the new engineer to have realistic expectations of their future workweek.

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Most popular questions from this chapter

A statistics instructor believes that fewer than 20\(\%\) of Evergreen Valley College (EVC) students attended the opening midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 of them attended the midnight showing. The Type Ierror is to conclude that the percent of EVC students who attended is _____ a. at least 20\(\%\) , when in fact, it is less than 20\(\% .\) b. \(20 \%,\) when in fact, it is 20\(\%\) . c. less than \(20 \%,\) when in fact, it is at least 20\(\%\) . d. less than \(20 \%,\) when in fact, it is less than 20\(\% .\)

Assume \(H_{0} : \mu \leq 6\) and \(H_{a} : \mu>6 .\) Is this a left-tailed, right-tailed, or two-tailed test?

Use the following information to answer the next seven exercises: Suppose that a recent article stated that the mean time spent in jail by a first-time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was three years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. Assume the distribution of the jail times is approximately normal. What symbol represents the random variable for this test?

You flip a coin and record whether it shows heads or tails. You know the probability of getting heads is 50%, but you think it is less for this particular coin. What type of test would you use?

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