FEV1, forced expiratory volume, the amount of air a person can blow out in 1 second after breathing in deeply, is an easy way to assess lung function. The units are liters/1 second.

Researchers studying asthma were interested in how children's FEV1 varied with their height.

Note:

- The data table is below

- The outcome variable is fev1

- The fev1 values are in liters of air in the rst second.

- The explanatory variable is height.

- The heights are measured in inches and there are 351 children in the dataset.

A. Produce a scatter plot with the fev1 observations and height, including the least squares regression line and place it here: (USE R: show the R function code and the output)

B. Based on this plot, does a line give a good summary of the relationship between FEV1 and height? Explain in detail your reasoning

C. Is the variability of the observed points around the tted least squares line roughly constant? Explain in detail your reasoning. Focus on the observations with heights between 50 and 65 inches, where there are good numbers of observations.

D. Explain why we do not have to look at the distribution of the standardized residuals to know that it is okay to use methods based on the t distribution to nd condence intervals and carry out hypothesis tests for these data.

E. No matter what you got for A through D, analyze the data using the least squares regression line.

Find the averages and standard deviations of fev1 and height and the Pearson correlation coefcient. Paste your results here:

F. Use the R lm function to t the least squares regression. (USE R: show the R function code and the output).

G. Calculate the slope estimate from the Pearson correlation coefcient and the standard deviations of fev1 and height. Show all intermediate steps and include the units for the slope.

H. Calculate the intercept estimate. Show all intermediate steps and include the units for the intercept.

I. Write out the equation for the tted regression line.

J. Explain why it is not of any concern that the least squares line has a negative value for the intercept.

K. The investigators want to know if these data provide evidence to support the hypothesis that mean FEV1 is higher for children who are taller. State the null and alternative hypotheses for the investigator's question.

L. Calculate the test statistic. Show all intermediate steps

M. What are the degrees of freedom for the test? Explain your reasoning and include your calculations.

N. Using your answer to question M explain how you know the P value is very small.

O. Using P < .01 as the criterion for statistical signicance, state the test conclusion in a sentence for the investigators.

P. If you are using R, you will need the anova function for the next question. (USE R: show the R function code and the output).

Q. Calculate R² from the sums of squares and interpret it.

R. Explain why it would not be wise to use your regression model to estimate the mean FEV1 of children 42 inches in height.

Data file name: fev_ages6_10

age	fev1	height	sex	smoke	female
9	1.708	57	female	no	1
8	1.724	67.5	female	no	1
7	1.72	54.5	female	no	1
9	1.558	53	male	no	0
9	1.895	57	male	no	0
8	2.336	61	female	no	1
6	1.919	58	female	no	1
6	1.415	56	female	no	1
8	1.987	58.5	female	no	1
9	1.942	60	female	no	1
6	1.602	53	female	no	1
8	1.735	54	male	no	0
8	2.193	58.5	female	no	1
8	2.118	60.5	male	no	0
8	2.258	58	male	no	0
7	1.932	53	male	no	0
6	1.878	53	female	no	1
9	2.352	59	male	no	0
9	2.604	61.5	male	no	0
7	2.578	62.5	male	no	0
9	2.988	65	female	no	1
9	2.348	60	male	no	0
8	2.98	60	female	no	1
9	2.1	60	female	no	1
9	3	65.5	male	no	0
8	2.673	60	female	no	1
7	2.093	57.5	female	no	1
8	2.175	59	female	no	1
9	2.725	59	male	no	0
8	2.071	55	male	no	0
8	1.547	57	male	no	0
8	2.004	57	male	no	0
9	3.135	60	female	no	1
8	2.42	59	male	no	0
8	1.931	57	female	no	1
9	2.076	57	female	no	1
7	1.624	54	male	no	0
8	1.344	52.5	female	no	1
6	1.65	55	male	no	0
8	2.732	60.5	male	no	0
9	2.797	61.5	female	no	1
9	3.556	62	male	no	0
8	1.703	54.5	male	no	0
6	1.634	54	male	no	0
9	2.57	57	male	no	0
9	3.016	62.5	female	no	1
7	2.419	60	female	no	1
8	1.698	57.5	female	no	1
8	2.123	60	male	no	0
8	2.481	60	female	no	1
6	1.481	51	female	no	1
8	1.94	59	male	no	0
6	1.747	57.5	male	no	0
9	2.069	58	male	no	0
7	1.631	55.5	female	no	1
9	2.56	60.5	female	no	1
8	1.962	57	male	no	0
8	2.531	58	female	no	1
9	2.715	60	male	no	0
9	2.457	59	male	no	0
9	2.09	59.5	male	no	0
7	1.789	56	male	no	0
9	3.842	69	male	no	0
6	1.719	53	female	no	1
7	2.111	57	female	no	1
6	1.695	53	female	no	1
8	2.211	63	male	no	0
8	1.794	54.5	male	no	0
7	1.917	58	female	no	1
8	2.144	63	female	no	1
7	1.253	52	male	no	0
9	2.659	61.5	male	no	0
9	2.126	62	male	no	0
9	3.029	61.5	female	no	1
9	2.964	64.5	male	no	0
7	1.611	57.5	male	no	0
8	2.215	60	female	no	1
8	2.388	60	female	no	1
9	2.196	61	male	no	0
9	1.751	58	male	no	0
9	2.165	61.5	male	no	0
7	1.682	55	male	no	0
8	1.523	55	male	no	0
8	1.292	52	female	no	1
7	1.649	54	male	no	0
9	2.588	63	male	no	0
9	2.574	60.5	female	no	1
6	1.979	56	male	no	0
8	2.354	58.5	male	no	0
6	1.718	55	male	no	0
7	1.742	58.5	female	no	1
7	1.603	51	female	no	1
8	2.639	59.5	female	no	1
7	1.829	54	female	no	1
7	2.084	58	male	no	0
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7	1.473	52.5	female	no	1
8	2.341	60.5	female	no	1
7	1.698	54.5	female	no	1
8	1.872	56.5	female	no	1
7	2.219	55	male	no	0
9	2.42	57	male	no	0
7	1.827	54.5	female	no	1
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6	1.338	53	male	no	0
8	2.09	57	male	no	0
8	1.697	59	female	no	1
8	1.562	55	male	no	0
9	2.04	55.5	female	no	1
7	1.609	51.5	female	no	1
8	2.458	61	female	no	1
9	2.65	63.5	male	no	0
8	1.429	57.5	male	no	0
8	1.675	53	male	no	0
9	1.947	56.5	female	no	1
8	2.069	54	male	no	0
6	1.572	52	male	no	0
6	1.348	53	male	no	0
8	2.288	61.5	female	no	1
9	1.773	58.5	male	no	0
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9	2.463	61	female	no	1
6	1.431	51	male	no	0
9	2.631	62	female	no	1
9	3.114	64.5	male	no	0
9	2.135	58.5	male	no	0
6	1.527	52.5	male	no	0
8	2.293	58	female	no	1
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8	2.927	63.5	male	no	0
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9	2.301	58.5	male	no	0
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9	2.592	60.5	female	no	1
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8	1.759	53	male	no	0
6	1.536	48	male	no	0
9	2.259	58.5	female	no	1
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9	2.571	60.5	male	no	0
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8	1.78	58.5	female	no	1
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9	2.893	64.5	male	no	0
6	1.713	50.5	male	no	0
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6	1.624	51.5	male	no	0
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9	3.004	64	female	no	1
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9	1.933	58	female	no	1
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9	2.316	59.5	female	no	1
9	1.606	57.5	female	no	1
7	1.165	47	male	no	0
6	2.102	55.5	female	no	1
9	2.32	57	female	no	1
9	2.23	61	male	no	0
9	1.716	55.5	male	no	0
7	1.79	53.5	male	no	0
8	2.187	61.5	female	no	1
9	2.717	61.5	male	no	0
7	1.796	55	male	no	0
9	1.953	58	male	yes	0
8	1.335	56.5	female	no	1
9	2.119	57	male	no	0
6	1.666	52	male	no	0
6	1.826	52.5	male	no	0
8	2.709	62.5	female	no	1
9	2.871	65	male	no	0
6	2.262	57.5	male	no	0
6	2.104	56.5	male	no	0
9	2.166	57.5	female	no	1
7	1.69	54	female	no	1
9	2.973	59.5	male	no	0
8	2.145	59.5	female	no	1
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6	1.697	55	female	no	1
9	2.455	60	female	no	1
7	1.92	56.5	male	no	0
9	2.164	60	male	no	0
9	2.13	59	female	no	1
8	2.993	63	female	no	1
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7	1.726	53	female	no	1
9	2.442	61.5	female	no	1
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8	2.305	64.5	female	no	1
9	1.969	59	female	no	1
8	1.556	58.5	female	no	1
9	2.042	62	male	no	0
8	1.512	53	female	no	1
6	1.423	49.5	male	no	0
9	3.681	68	male	no	0
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6	1.796	55	female	no	1
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7	1.415	53.5	female	no	1
9	2.076	60.5	male	no	0
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9	1.886	56	female	no	1
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