Exercises 20.46-20.58 require the use of a computer and...

Exercises $20.46-20.58$ require the use of a computer and software. Use a $5 \%$ significance level unless specified otherwise. Refer to Example 20.7. Suppose that the responses were recoded so that the numbers equalled the midpoint of the range of percentiles; that is: $97.5=$ The candidate is in the top $5 \%$ of applicants. $92.5=$ The candidate is in the top $10 \%$ of applicants, but not in the top $5 \%$. $82.5=$ The candidate is in the top $25 \%$ of applicants, but not in the top $10 \%$. $62.5=$ The candidate is in the top $50 \%$ of applicants, but not in the top $25 \%$. $25=$ The candidate is in the bottom $50 \%$ of applicants. Can we conclude that differences exist between the ratings assigned by the four managers? b Compare your answer in part (a) with the one obtained in Example 20.7. Are they the same? Explain why.

Exercises $20.46-20.58$ require the use of a computer and software. Use a $5 \%$ significance level unless specified otherwise.
Refer to Example 20.7. Suppose that the responses were recoded so that the numbers equalled the midpoint of the range of percentiles; that is:
$97.5=$ The candidate is in the top $5 \%$ of applicants.
$92.5=$ The candidate is in the top $10 \%$ of applicants, but not in the top $5 \%$.
$82.5=$ The candidate is in the top $25 \%$ of applicants, but not in the top $10 \%$.
$62.5=$ The candidate is in the top $50 \%$ of applicants, but not in the top $25 \%$.
$25=$ The candidate is in the bottom $50 \%$ of applicants.
Can we conclude that differences exist between the ratings assigned by the four managers?
b Compare your answer in part (a) with the one obtained in Example 20.7. Are they the same? Explain why.

Key Concepts

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of three or more groups to determine if at least one group mean is significantly different from the others. In problems where multiple managers or raters provide evaluations, ANOVA tests whether the differences in their ratings are statistically significant, thereby helping to assess consistency or bias among the evaluators.

Hypothesis Testing and Significance Level

Hypothesis testing involves formulating a null hypothesis that there is no difference between groups and an alternative hypothesis that there is a difference. The significance level, such as 5%, defines the threshold for rejecting the null hypothesis. This approach provides a systematic method for decision-making in studies comparing group ratings or measurements.

Data Transformation and Recoding

Data transformation, such as recoding ratings to represent midpoints of percentile ranges, is a process used to convert qualitative or ordinal data into a numerical form that can be analyzed statistically. This transformation can make the data conform more closely to the assumptions of parametric tests and may alter the interpretation of the results if the scale properties are changed.

Robustness and Invariance of Statistical Procedures

Statistical robustness and invariance refer to the ability of a test to yield similar conclusions even when the data undergo certain transformations. Comparing results from different coding schemes checks whether the conclusions drawn about differences among groups (such as ratings by managers) are consistent, demonstrating the reliability of the statistical method used.

Transcript

00:01 What's up, stat katz? so this is question 22.

00:03 It's probably the most sexist question i've ever come across in a textbook.

00:09 Factory hiring people to work on an assembly line gives job applicants a test of manual agility.

00:16 This test counts how many strangely shaped pegs that applicant can fit into matching holes in a one -minute period.

00:25 The table below summarizes the data by sex of the job applicant, assume that.

00:30 That all conditions necessary for inference are met.

00:33 All right, so this is what they give us.

00:36 So we have sample size of 50 each, and they tested men and women against each other to see how fast, how many pegs they could put into a hole in a minute.

00:51 So question a is, find 95 % confidence intervals for the average number of pegs that males and females can each place.

01:03 So we're not doing a two -tailed interval.

01:06 We're doing, i'm sorry, a two -sample interval.

01:10 We're doing a one sample interval, and we're doing them separately.

01:13 So let's go ahead and grab our calculator.

01:15 We're going to hit the stat button.

01:17 We're going to go over to tests, and then we're going to do just plain t interval, which is number eight for a shortcut.

01:26 So let's do men first.

01:29 So the average was 19 .39.

01:34 Standard deviation to .52.

01:38 Sample size 50, 95%.

01:43 Yep.

01:43 Okay, let's calculate that.

01:46 So we have 18 .674 to 20 .106.

01:52 So let's do the females now.

01:55 And then we'll go ahead and copy those over into our whiteboard.

01:58 So same thing, just with the female information to calculate.

02:12 All right, 16 .947 to 18 .873.

02:18 So let's go ahead and mark those down.

02:25 So the male is 18.

02:33 Okay, so that's for the men, women.

02:48 Okay, so these give us the average amount of pegs that men and women can place into the holes in one minute.

03:00 And this is an estimate of the population.

03:04 So we can't compare these to each other.

03:07 So these are just, this is an interval for the average number of pegs that men can put in a hole in one minute.

03:16 And this is the average number of pegs that females can put in a hole in one minute.

03:22 So part b, what is the interval? suggest? well, the interval suggests that men are slightly faster at putting pegs in the holes.

03:36 I don't want to say better.

03:37 I'm just going to say faster at doing it.

03:41 It also says, so those intervals interlap.

03:45 What does this suggest about any sex -based difference in manual agility? so, yeah, they do overlap a little bit.

03:52 We have 18 .873 and 18 .674.

03:55 So that suggests that there isn't a significant difference between men and women, but again, these are estimates of the population.

04:09 So they do overlap, so i don't want to say there's a difference in the sexes.

04:20 So i'm going to say non -significance because they do overlap right here.

04:32 So the men and women are sharing a common average for their specific populations...

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