Question

Children's height as a function of their age has been researched so extensively that we can consider known results to describe the relationship for all children in the United States. For instance, between the ages of 13 and 15, population mean height for teenage males (in inches) satisfies

μ_y = 22 + 3x,

where x is age in years. Spread about the line is 3.1 inches.

(a) Notice that the slope of the regression line for the population is β₁ = 3. If we were to take repeated random samples of 25 males between the ages of 13 and 15 and regress their heights on their ages, then the slopes b₁ would vary from sample to sample. At what slope value would their distribution be centered? (Answer as a whole number.)

(b) If we were to take repeated random samples of males between the ages of 13 and 15 and regress their heights on their ages, then the slopes b₁ would vary from sample to sample. In which case would the distribution of sample slopes have less spread?

when sampling 25 males

when sampling 10 males       

(c) What notation is used for the spread about the line (3.1 inches)?

σ

s     

(d) On average, how much shorter do you predict a 13-year-old to be compared to a 15-year-old? (Answer as a whole number.)
in
(e) The linear regression model does a good job of summarizing the relationship between height and age for males in a particular age range, such as between 13 and 15 years old. Which two conditions would not be met if we attempted to perform inference about the height/age relationship based on a random sample of 250 males all the way from newborn to 25 years old?

Explanatory/response values should constitute a random sample of independent pairs.

Scatterplot should appear linear.

Spread of responses should appear fairly constant over the range of explanatory values.

Sample size should be large enough to offset non-normality in responses.

Answer 1

Solutin :

We are given childrens height as a function of their age, between the ages of 13 and 15, population mean height for teenage satisfies the linear equation as

mean = 22 + 3x,   where x is age in years.

a) At the slope value of 0 Their distribution is centered.

Explanation:

Since it is given that the ages can take value 13,14 and 15, and also the sample size is 25, so it is possible the one value is greater than the other two, so for the distribution to be centered the slope should be 0.

b) Statement A is correct.i.e.

When sampling 25 male

Explanation:

Since if the sample size is large then the spred is less.

c) Option b is correct. i.e s

The sample spread is denoted by s.

d) the mean height when age is 13 is = 22 + 3*13 = 61

The mean height when the age is 15 = 22 + 3*15 = 67

Therefore, 67 - 61 = 6

Hence, we predict 6 inch shorter on an average.

e) Option B and C is correct choices.

Explanation:

Since we fit model on ages between 13 to 15 but here we take ages 0 to 15 , so linearity may affected and since linearity is affected therefore spread of responses should not be constant over the range of explanatory variable.

.

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