Researchers thought that right handed people perform better...

Researchers thought that right-handed people perform better with their right hands compared to their left hands. A video game designed to test hand-eye coordination was used to measure performance. The time taken (in seconds) to complete a task requiring hand-eye coordination was recorded. The subjects were 25 right-handed individuals who were familiar with using computers. The data is attached to this email. Suppose the parameter is the average difference (left-hand time minus right-hand time) in completion times for all right-handed individuals who are familiar with using computers. (a) Please write down the research hypothesis using the correct notation. (b) Please use Minitab to find the test statistic and p-value. Suppose you ask the researchers about the selection process for the 25 people in the sample. The researchers explain that they recruited their subjects from a large group of gamers. (c) Which one of the following problems is most likely to have occurred? - Responder bias - Interviewer bias - Non-response bias - Response bias - Under-coverage - Non-response bias - Random fluctuation - Standard error - Random variation - Margin of error (d) Please explain briefly and in simple words how the problem identified above occurred.

Transcript

00:03 So we have some data which comes from a study of americans from a variety of professions and they were asked they consider themselves left -handed, right -handed, or ambidextrous.

00:15 And this column is full of the professions, but we only really care about these two.

00:20 We have the total here.

00:23 And for the first part, we're asked to see if there's enough evidence from this data to show that more than 10 % of those in these professions are these left -handed control.

00:35 Percent of the relation is left -handed.

00:41 So for right now we only care about this number, 67 and 635.

00:49 So our null hypothesis and the alternative hypotheses are going to be p is equal to 0 .10 and p is greater than 0 .10 respectively for the null and alternative.

01:03 And we are going to set the alpha to be 0 .05, and the proportion of left -handed people in the sample within these professions is 0 .15, and we get that from doing 67 divided by 635.

01:27 And we're going to use a one proportion z test for this, and the formula is given as z equals p hat.

01:40 Minus p divided by, and this is the set p value, this will be our p of 0 .10, divided by the square root of p times 1 minus p, all over n.

01:55 So what that means for us is p hat, which is the 667 divided by 635 minus 0 .1 divided by the square root of 0 .1 times 1 minus 0 .1 all over 635.

02:29 And then the formula we get, or the value we get is 0 .4629.

02:37 And the p value is this.

02:41 I'll show how we get it.

02:43 So this z score ends up being 0 .463, we'll say.

02:50 And then we do a table lookup, or like me, i use my spreadsheet, and you find the area to the left of it, which is 0 .678.

02:58 But we don't want that.

02:59 We want one minus that value.

03:01 We want to basically find the area above it.

03:02 So in a picture, what i just said was, here's our distribution.

03:11 Here's the z score of zero.

03:12 We found the z score of 0 .463.

03:18 We want to find this area to the right.

03:21 But tables give us the error to the left.

03:23 So we just do one minus that value.

03:24 So we find the area to be 0 .32.

03:30 And so with, since that p value does not fall below our alpha, we fail to reject.

03:45 And so that means we don't have evidence to show, to suggest that the more than 10 % of these professions are left -handed.

03:59 Right.

04:01 And then the next question asks us to make a 99 % confidence interval for the proportion of americans were ambidextrous.

04:09 So for that, we need to find the proportion of people that are ambidextrous, which is 32 out of the 635.

04:25 So the proportion of ambidextrous.

04:29 32 and 635, which ends up being, let's show the whole thing here.

04:40 We'll get to the calculations in a moment.

04:44 0 .05 -03, we'll just round it to 0 .05.

05:00 So our confidence interval, because it's based on proportion, is given with the following formula, p hat plus minus z alpha over two, multiplied by p hat times one minus p hat over n there we go so we find the z value it's a pretty standard z value it's 2 .576 and the p hat we already have so we just plug our values into our formula and i called this part out this this square root term is called the or factors called the standard error and z times the standard error that's also called the margin of error and i usually like to call that out when i'm making a confidence in all just so i know how far above and below the mean we're going so when we substitute in our values what we end up with is the following straightforward formula it's 0 .05 plus minus and then the margin of error ends up being 0 .022.

06:15 And that gets us the, so we, plus minus, sorry, the confidence interval of 0 .028 up to 0 .073.

06:40 And that's if we, well, i took the precision of my spreadsheet, but if we use the values as i have in the approximating.

06:51 Here.

06:53 We'll end up being this.

06:54 So that's our 99 % confidence interval for the proportion of americans who are ambidextrous, which is based on this sample.

07:07 All right.

07:08 And then the third part of this question asks us to determine if there's enough evidence to conclude that the proportion of left -handed orthopedic surgeons is less than the proportion of left -handed architects.

07:23 So that's where these architects and its architect and the orthopedic surgeons, data come into play.

07:30 So our null and alternative hypotheses are the following.

07:37 I'll come down to this part and i'll show you what all this stuff is in a moment.

07:43 So our null and alternative hypotheses are listed here.

07:47 We have the nulls that the proportion of the architects.

07:51 I'm going to make this bigger actually.

08:03 The proportion of architects is equal to the proportion of orthopedic surgeon and this is the proportion of left -handed architect.

08:12 And orthopedic surgeons, that is.

08:14 And the alternative hypothesis is that the orthopedic surgeon's proportion is less than the proportion of left -handed architects.

08:25 So the way we do this is we do a two -proportionation z test, and we're going to set the alpha to be, we'll do 0 .05 again.

08:44 That's a pretty good one.

08:47 And our formula is the following.

08:49 Z equals p hat 1 minus p hat 2.

08:54 And p hat 1, p .at 2.

08:55 Those are just going to be our proportions of the architect and orthopedic surgeons divided by the square root of the pooled proportion times 1 minus the pooled proportion.

09:10 And i'll show you how we get that in the second.

09:12 Divided by 1 over sample size of 1 plus 1 over sample size of 2...

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