The plot below is a two way scatter plot of head circumference (cm) vs gestation

Question

The plot below is a two way scatter plot of head circumference (cm) vs gestational age (weeks) for a sample of 100 low birthweight infants, and the output from fitting a simple linear regression model to the data, regressing headcirc (Y) on gestage(X).

Analysis of Variance

Sum of Mean
Source DF Squares Square F Value Pr > F

Model 1 386.86737 386.86737 152.95 <.0001
Error 98 247.88263 2.52941
Corrected Total 99 634.75000


Root MSE 1.59041 R-Square 0.6095


Parameter Estimates

Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|

Intercept Intercept 1 3.91426 1.82915 2.14 0.0348
gestage gestage 1 0.78005 0.06307 12.37 <.0001

a) Suppose that a new newborn is selected from the underlying population of low birthweight infants with gestational age of 29 weeks. Give a 95% prediction interval for the new value of head circumference.

b) Explain the difference between a prediction interval and a confidence interval.

Explanation / Answer

alpha = 0.05

alpha/2 = 0.025

df= 99

t = 1.987

I: 3.91 +/- 1.987 * 2.14

3.91 +/- 4.25218

-0.34218 < miu < 8.16218

b)

Confidence intervals tell you about how well you have determined the mean. Assume that the data really are randomly sampled from a Gaussian distribution. If you do this many times, and calculate a confidence interval of the mean from each sample, you'd expect about 95 % of those intervals to include the true value of the population mean. The key point is that the confidence interval tells you about the likely location of the true population parameter.

Prediction intervals tell you where you can expect to see the next data point sampled. Assume that the data really are randomly sampled from a Gaussian distribution. Collect a sample of data and calculate a prediction interval. Then sample one more value from the population. If you do this many times, you'd expect that next value to lie within that prediction interval in 95% of the samples.The key point is that the prediction interval tells you about the distribution of values, not the uncertainty in determining the population mean.

Prediction intervals must account for both the uncertainty in knowing the value of the population mean, plus data scatter. So a prediction interval is always wider than a confidence interval.

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