Question

As part of a research program for a new cholesterol drug, a pharmaceutical company would like to investigate the relationship between the ages and LDL (low-density lipoprotein) cholesterol of men. The following data set shows the ages and LDL cholesterol levels of seven randomly selected men. Construct a 95% confidence interval to estimate the average LDL cholesterol level of a 31-year-old man. Click the icon to view the datatable Determine the upper and lower limits of the confidence interval UCL- LCL (Round to one decimal place as needed.)

Answer 1

Solution

Back-up Theory

100(1 - α)% Confidence Interval (CI) for y_cap at x = x₀ is:

(y_cap at x = x₀) ± t_{n – 2, α/2}xs√[(1/n) + {(x₀ – Xbar)²/Sxx}] .................................................................................................... (1)

Where

Estimate of σ² is given by s² = (Syy – β_1cap²Sxx)/(n - 2) ………............................….....................................................…..(2)

[Mean X = Xbar = (1/n) Σ(i = 1 to n)x_i ; Mean Y = Ybar = (1/n) Σ(i = 1 to n)y_i Sxx = Σ(i = 1 to n)(x_i – Xbar)² Syy = Σ(i = 1 to n)(y_i – Ybar)² ; Sxy = Σ(i = 1 to n){(x_i – Xbar)(y_i – Ybar)} ].......................................................................... (6)

Now to work out the solution,

Final answer in the stipulated format is given below. Details of calculations follow at the end.

95% confidence interval for the LDL of a 31 year old man is:

UCL = 190.1

LCL = 137.0

Answer

Details of calculations

Let x and y respectively represent the age and LDL.

Data

i	xi	yi
1	24	140
2	36	177
3	27	156
4	31	199
5	42	157
6	32	142
7	42	210

Calculations

n	7
Xbar	33.43
ybar	168.71
Sxx	291.71
Syy	4527.43
Sxy	591.86
β1hat	2.0288932
β0hat	100.8913
s2	665.3227
s	25.7939
α	0.05
n-2	5
tn-2,α/2	2.5706
x0	31
yhat at x0	163.7870
CIYhatLB	137.0112
CIYhatUB	190.5627

DONE