Question

Chapter 6- Question # 2 e, f

Researchers have collected data from a random sample of six students on the number of hours spent studying for an exam and the grade received on the exam as given in Table 6.5.

Table 6.5

Observation	Grade	Number of Hours Studying
1	85	8
2	73	10
3	95	13
4	77	5
5	68	2
6	95	12

e) Find and interpret a 99% confidence interval for the predicted grade for an individual who spends 10 hours studying.

f) Find and interpret a 99% confidence interval for the mean grade of all individuals who spend 10 hours studying.

Answer 1

Based on the given data,

e) A 95% Confidence interval for the predicted grade for an individual who spends 10 hours studying. can be determined using the formula:

where = Predicted grade for an individual who spends 10 hours studying.    s = Standard error of the estimate =    t = critical value of t for n - 2 = 6 - 2 = 4 degrees of freedom

To find , we first need to run a linear regression model and obtain the fitted regression line:

= a + bx

where,

X = predictor / independent variable (Study Hours) Y = Response / dependent variable (Grade)

Substituting the values,

= 2.213

a = 82.167 - 2.213 ( 50 )

= 63.728

Hence, = 63.728 + 2.213X

Computing for each value of X,

For X= 10,   = 63.728 + 2.213 (10) = 85.854

s = Standard error of the estimate =

= 7.27081

  

= 89.333

And, t critical value at 1% level for 4 degrees of freedom,

Substituting the value in the 99% confidence interval,

= (71.4435,100.2654)

Hence, 99% confidence interval for the predicted grade for an individual who spends 10 hours studying is (71.4435,100.2654). It implies that if 100 such repeated samples are collected and the value is predicted as for the above sample, the interval (71.4435,100.2654) would contain the true grade value 99 out of 100 times.

b. To obtain the 95% confidence interval, we may use the same formula as above, with the only change being the significance level for the critical value t at 5% level for 4 df:

Substituting the values,

= (77.1641,94.5448)

Hence, 95% confidence interval for the predicted grade for an individual who spends 10 hours studying is (77.1641,94.5448). It implies that if 100 such repeated samples are collected and the value is predicted as for the above sample, the interval (77.1641,94.5448) would contain the true grade value 95 out of 100 times.

As the level of confidence decreases, the confidence interval gets narrower.