Scores in the first and fourth (final) rounds for a sample...

Scores in the first and fourth (final) rounds for a sample of 20 golfers who competed in PGA tournaments are shown in the following table. Suppose you would like to determine if the mean score for the first round of a PGA Tour event is significantly different than the mean score for the fourth and final round. Does the pressure of playing in the final round cause scores to go up? Or does the increased player concentration cause scores to come down? a. Use $\alpha=.10$ to test for a statistically significantly difference between the population means for first- and fourth-round scores. What is the $p$-value? What is your conclusion? b. What is the point estimate of the difference between the two population means? For which round is the population mean score lower? c. What is the margin of error for a $90 \%$ confidence interval estimate for the difference between the population means? Could this confidence interval have been used to test the hypothesis in part (a)? Explain.

Scores in the first and fourth (final) rounds for a sample of 20 golfers who competed in PGA tournaments are shown in the following table. Suppose you would like to determine if the mean score for the first round of a PGA Tour event is significantly different than the mean score for the fourth and final round. Does the pressure of playing in the final round cause scores to go up? Or does the increased player concentration cause scores to come down?
a. Use $\alpha=.10$ to test for a statistically significantly difference between the population means for first- and fourth-round scores. What is the $p$-value? What is your conclusion?
b. What is the point estimate of the difference between the two population means? For which round is the population mean score lower?
c. What is the margin of error for a $90 \%$ confidence interval estimate for the difference between the population means? Could this confidence interval have been used to test the hypothesis in part (a)? Explain.

Key Concepts

Hypothesis Testing

This is the statistical procedure used to decide whether the observed sample evidence is strong enough to infer that a particular condition holds for the entire population. In the context of comparing mean scores, the hypothesis test determines if there is enough evidence to argue that the population means for the two rounds are different, using a predetermined significance level (such as ? = 0.10).

Paired Sample t-Test

This is a statistical test used when comparing two related samples or paired observations, such as scores from the same subjects under two different conditions. It accounts for the fact that the observations are not independent and evaluates whether the mean difference between the pairs is significantly different from zero.

p-Value

The p-value is a probability that measures the strength of the evidence against the null hypothesis. It indicates the likelihood of observing the sample data, or something more extreme, if the null hypothesis is true. A p-value lower than the chosen significance level (?) suggests that the null hypothesis should be rejected.

Point Estimate

A point estimate provides a single best guess of a population parameter, such as the difference between two means. In comparative studies, the point estimate is typically calculated as the difference between the two sample means and gives an indication of which group (or condition) has the higher or lower average score.

Confidence Interval

A confidence interval gives a range of values within which the true population parameter is likely to lie, with a certain level of confidence such as 90%. It provides a measure of uncertainty around the point estimate and can be used for inference; if the interval does not contain a value of interest (like zero in a difference test), one can draw conclusions about the significance of the result.

Margin of Error

The margin of error is the amount added and subtracted from the point estimate to form the confidence interval. It reflects the maximum expected sampling error and determines the precision of the estimate. A smaller margin of error implies a more precise estimate of the population parameter.

Transcript

00:01 I'm using the ti -84 to do the analysis on this question.

00:04 And in my calculator, i put in list one the first round scores, and in list two, i put the fourth round scores.

00:11 And then in list three, i'm having the difference.

00:15 And i'm subtracting and having the difference be first round minus the second round.

00:21 And it doesn't matter if i had subtracted the other way.

00:23 But you do want to label that down someplace, which way you're subtracting.

00:26 We will be assuming that the mean difference is zero.

00:30 That there is no difference between the scores of the first and fourth final round, and alternately that there is a difference.

00:37 It will analyze those differences if there are any shortly.

00:40 And we're using an alpha level of 0 .10, so 10 % significance level.

00:47 And when we do the calculations on part a, we find that the x bar for those differences is equal to negative 1 .05 with a sample standard deviation of 3 .3162, and we know our sample size is 12.

01:04 So our test statistic, our t value, which will have, oh, and i said our sample size is 12, our sample size is 20.

01:12 My previous question was 12.

01:15 And so our degrees of freedom is 19, and we'll take that difference that we got minus the difference we're assuming, the sample standard deviation, divided by the square root of 20, and that test statistic is negative 1 .41, and it's going to round to 6.

01:32 And so the p value associated with that is that likelihood of having this lower tail and we'll double it to have the opposite's upper tail.

01:43 And this comes out to be 0 .1730.

01:47 And so we're using a 10 % significance level, and this is higher than the 10 % significance level.

01:54 So we fail to reject the null.

02:00 So there is not evidence of a difference, not evidence of a difference.

02:09 Now, in part b, we're asked to look at, give point estimates for what the difference is.

02:19 And our estimate was that 1 .05.

02:23 And it appears as though, i mean, we can find exactly what those population means, but the difference between the mean of the first round minus the mean of the second round is equal to negative 1 .05, meaning that this score is actually higher.

02:44 And we can see that by actually calculated.

02:47 I'm stat calculate one variable statistics, and the second round has a mean of 70 .7.

02:55 That's for round two, or round four, actually...

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