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Chapter 20: Problem 56

In Exercise 54 , we saw a 98\% confidence interval of (-40,-22) minutes for \(\mu_{\text {Top }}-\mu_{\text {Front }}\) the difference in time it takes top- loading and front-loading washers to do a load of clothes. Explain why you think each of the following statements is true or false: a. \(98 \%\) of top loaders are 22 to 40 minutes faster than front loaders. b. If I choose the laundromat's top loader, there's a \(98 \%\) chance that my clothes will be done faster than if I had chosen the front loader. C. If I tried more samples of both kinds of washing machines, in about \(98 \%\) of these samples l'd expect the top loaders to be an average of 22 to 40 minutes faster. d. If I tried more samples, l'd expect about \(98 \%\) of the resulting confidence intervals to include the true difference in mean cycle time for the two types of washing machines. e. I'm \(98 \%\) confident that top loaders wash clothes an average of 22 to 40 minutes faster than front-loading machines.

Short Answer

Expert verified
Statements a, b and c are false while statements d and e are correct interpretations of the given confidence interval.

Step by step solution

01

Understanding confidence intervals

A confidence interval estimates the likely range of values for a population parameter, calculated from a sample data. It is not about the chances of individual observations falling in that range. So the confidence interval of (-40,-22) for the difference in time it takes top-loading and front-loading washers implies that we are 98\% confident that the true difference of means lies within this interval, not that 98\% of observations will fall in this range.
02

Evaluating statement a

The statement is incorrect. The confidence interval does not imply that 98\% of top loaders are 22 to 40 minutes faster than front loaders. It's about the difference in means not individual observations.
03

Evaluating statement b

The statement is false. The confidence interval does not provide the probability of one single event, in this case, if clothes will be done faster in top loader.
04

Evaluating statement c

This statement is also false. The confidence interval cannot predict about the percentage of samples that would fall in a certain range.
05

Evaluating statement d

This statement is true. This is essentially the definition of a confidence interval - in repeated sampling, we'd expect about 98% of the resulting confidence intervals to include the true mean difference.
06

Evaluating statement e

The statement is true since it directly refers to the interpretation of the confidence interval. The confidence interval of (-40,-22) does suggest that we are 98% confident that top loaders wash clothes an average of 22 to 40 minutes faster than front-loading machines.

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Key Concepts

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

Population Parameter
The population parameter in statistics represents a numerical characteristic of a population, like its mean or standard deviation. In the context of the given exercise, it refers to the actual average difference in cycle time between top-loading and front-loading washers. This parameter is fixed but generally unknown, as it represents the true average for all the machines that exist, not just the ones sampled.

When conducting statistical analysis, researchers are usually interested in estimating these population parameters because they want to draw conclusions about the entire population. However, because it's typically impossible to survey every single member of a population, statisticians use sample data to make inferences. This is where confidence intervals come into play. They offer a range within which the parameter likely falls.
Sample Data
Sample data is a subset of the population that researchers study to draw conclusions about the entire population. In the exercise, researchers collected data from a certain number of top-loading and front-loading washers to analyze their differences in cycle time. This data is then used to calculate the confidence interval.

The quality of the sample is crucial, as it should adequately represent the population to provide reliable results. If the sample is biased or too small, it may not reflect the true characteristics of the population, leading to inaccurate inferences. When sample data is used correctly, it can help estimate the population parameter accurately.
Mean Difference
The mean difference refers to the difference between the average values of two groups. In the context of the exercise, it is the average difference in cycle time between top-loading and front-loading washers. This mean difference is a pivotal aspect of statistical analysis when comparing two groups.

In the exercise, the confidence interval of (-40, -22) minutes suggests that the average time taken by front-loading washers is between 22 to 40 minutes longer than that of top-loading washers. It's important to note that this interval offers a range where this mean difference is likely to lie, not a definitive value. Understanding mean difference is crucial for grasping how two populations compare to each other and helps make informed decisions based on data.
Statistical Inference
Statistical inference encompasses the methods used to make conclusions about a population based on sample data. It involves several processes like hypothesis testing, estimating population parameters, and creating confidence intervals.

In this exercise, statistical inference is applied in determining the confidence interval for the difference in cycle times between the two types of washers. The confidence interval provides an estimation for where the true mean difference lies with a certain level of confidence. This helps to understand if the observed differences in the sample can be generalized to the entire population of washers.
  • It involves using sample data to make educated guesses about the population.
  • Helps assess the reliability of those guesses using statistical measures like confidence intervals and significance levels.
By grasping statistical inference, students can better interpret data and make decisions based on their analyses.

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Most popular questions from this chapter

In the same article from Exercise 46, Time magazine, reporting on a survey of men's attitudes, noted that "Young men are more comfortable than older men talking about their problems." The survey reported that 80 of 129 surveyed 18 - to 24 -year-old men and 98 of 18425 - to 34-year-old men said they were comfortable. What do you think? Is Time's interpretation justified by these numbers?

The Journal of the American Medical Association reported a study examining the possible impact of air pollution caused by the \(9 / 11\) attack on New York's World Trade Center on the weight of babies. Researchers found that \(8 \%\) of 182 babies born to mothers who were exposed to heavy doses of soot and ash on September 11 were classified as having low birthweight. Only \(4 \%\) of 2300 babies born in another New York City hospital whose mothers had not been near the site of the disaster were similarly classified. Does this indicate a possibility that air pollution might be linked to a significantly higher proportion of low-weight babies? a. Test an appropriate hypothesis at \(\alpha=0.10\) and state your conclusion. b. If you concluded there is a difference, estimate that difference with a confidence interval and interpret that interval in context.

The global survey we learned about in Exercise 49 also asked respondents how they felt about the statement "I try to avoid eating fast foods." The random sample of 800 included 411 people 35 years old or younger, and of those, 197 agreed (completely or somewhat) with the statement. Of the 389 people over 35 years old, 246 people agreed with the statement. Is there evidence that the percentage of people avoiding fast food is different in the two age groups?

The Core Plus Mathematics Project (CPMP) is an innovative approach to teaching Mathematics that engages students in group investigations and mathematical modeling. After field tests in 36 high schools over a three-year period, researchers compared the performances of CPMP students with those taught using a traditional curriculum. In one test, students had to solve applied algebra problems using calculators. Scores for 320 CPMP students were compared to those of a control group of 273 students in a traditional math program. Computer software was used to create a confidence interval for the difference in mean scores. (Journal for Research in Mathematics Education, 31, no. 3) Conf level: \(95 \%\) Variable: Mu(CPMP) - Mu(CtrI) Interval: (5.573,11.427) a. What's the margin of error for this confidence interval? b. If we had created a \(98 \% \mathrm{Cl}\), would the margin of error be larger or smaller? c. Explain what the calculated interval means in this context. d. Does this result suggest that students who learn mathematics with CPMP will have significantly higher mean scores in algebra than those in traditional programs? Explain.

Researchers comparing the effectiveness of two pain medications randomly selected a group of patients who had been complaining of a certain kind of joint pain. They randomly divided these people into two groups, then administered the pain killers. Of the 112 people in the group who received medication A, 84 said this pain reliever was effective. Of the 108 people in the other group, 66 reported that pain reliever B was effective. a. Write a \(95 \%\) confidence interval for the percent of people who may get relief from this kind of joint pain by using medication A. Interpret your interval. b. Write a \(95 \%\) confidence interval for the percent of people who may get relief by using medication B. Interpret your interval. c. Do the intervals for \(A\) and B overlap? What do you think this means about the comparative effectiveness of these medications? d. Find a \(95 \%\) confidence interval for the difference in the proportions of people who may find these medications effective. Interpret your interval. e. Does this interval contain zero? What does that mean? f. Why do the results in parts \(c\) and e seem contradictory? If we want to compare the effectiveness of these two pain relievers, which is the correct approach? Why?
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