Question

a. all Hypothesis Tests must include all four steps, clearly labeled;

b. all Confidence Intervals must include all output as well as the CI itself

c. include which calculator function you used for each problem.

3. The authors of the paper “Statistical Methods for Assessing Agreement Between Two Methods of Clinical Measurement” (International Journal of Nursing Studies [2010]: 931-936) compared two different instruments for measuring a person’s ability to breathe out air. (This measurement is helpful in diagnosing various lung disorders.) The two instruments considered were a Wright peak flow meter and a mini-Wright peak flow meter. Seventeen people were randomly selected to participate in the study, and for each person air flow was measured once using the Wright meter and once using the mini-Wright meter.

Mini-Wright Meter (x)	Wright Meter (y)
512	494
430	395
520	516
364	413
380	442
658	650
445	433
432	417
626	656
428	434
500	476
600	557
260	267
477	478
259	178
350	423
451	427

(this table is color-coded for a reason; please pay close attention to it)

Suppose that the Wright meter is considered to provide a better measure of air flow, but the mini-Wright meter is easier to transport and to use. If the two types of meters produce different readings but there is a strong relationship between the readings, it would be possible to use a reading from the mini-Wright meter to predict the reading that the larger Wright meter would have given.

a. Calculate and interpret the value of the correlation coefficient. In your interpretation, you should answer the question: “Based

on the data, would you be comfortable using the mini-Wright meter reading to predict the larger Wright meter reading?”

b. Run the hypothesis test (at α = 5%) to determine if there is a correlation between the Wright meter and the mini-Wright meter.

c. Use the given data to find an equation to predict the Wright meter reading (y) using a reading from the mini-Wright meter (x).

d. What would you predict for the Wright meter reading for a person whose mini-Wright meter reading was:

600?

500?

200?

Do any of these predictions pose any problems? Explain.

Answer 1

a). Correlation coefficient:

where

Let us form a table as below:

	x	y	x^2	y^2	x*y
	512	494	262144	244036	252928
	430	395	184900	156025	169850
	520	516	270400	266256	268320
	364	413	132496	170569	150332
	380	442	144400	195364	167960
	658	650	432964	422500	427700
	445	433	198025	187489	192685
	432	417	186624	173889	180144
	626	656	391876	430336	410656
	428	434	183184	188356	185752
	500	476	250000	226576	238000
	600	557	360000	310249	334200
	260	267	67600	71289	69420
	477	478	227529	228484	228006
	259	178	67081	31684	46102
	350	423	122500	178929	148050
	451	427	203401	182329	192577
Total	7692	7656	3685124	3664360	3662682

The correlation coefficent is 0.9433 which is very high. Therefore,

Based on the data, I will be comfortable using the mini-Wright meter reading to predict the larger Wright meter reading?”

b. Null Hypothesis: There is no correlation or the correlation is absent.

Alternate Hypothesis: The correlation is present

Level of significance:

Test statistic: will have t distribution with n-2 df.

The critical value of t(15,5%) is 2.1314. Since, the t calculated value is >tcritical, we reject the Null hypotheis and conclude that the correlation coefficient is significant.

c. Use the given data to find an equation to predict the Wright meter reading (y) using a reading from the mini-Wright meter (x).

we just need two more quantities:

and

The estimate of slope

The estimate of intercept:

  

The estimated regression equation is

d. What would you predict for the Wright meter reading for a person whose mini-Wright meter reading was:

When x=600, y=11.4817+0.9699*600=593.4217

x=500, y=11.4817+0.9699*500=496.4317

x=200,y=11.4817+0.9699*200=205.4617

It can be observed that in our data when x=600, the observed value of y=557 and the estimated value is 593

  x=500, the observed value of y=476 and the estimated value is 496

There are diffferences between actual and estimated values.