Question

In ANOVA test, acceptance of the null hypothesis indicates that:

1. The slope of the SLR model is null

1. True                 b)   False        

A manufacturer produces ball bearings, normally distributed, with unknown mean and standard deviation. A sample of 25 has a mean of 2.5cm. The 99% confidence interval has length 4cm (double-sided).

2. Which statement is correct (use t-distribution):

1. s² = 23.42cm²        b) s² = 12.82cm²              c)   s= 3.58cm                   d) s= 4.84cm     

3. The 99% prediction interval measures (use t-distribution):

a) 20.38 cm                  b) 10.21 cm                      c) 20.42 cm                               d) 10.19 cm

Answer 1

Suppose, we intend to fit a simple linear regression model, by regressing the response y on predictor x, with the fitted regression expressed as:

where are the estimated intercept and slope coefficient.

Here, the slope coefficient gives the strength and direction of the causal relationship between the two, by measuring the change in response y for a unit change in predictor x. Hence, a slope equal to zero implies no causal relationship between x and y.

Using regression, we test whether a significant causal relationship between x and y; and the negation of this statement would be the null hypothesis - No causal relationship between x and y.

To test: Vs   

Hence, the correct option would be:

Acceptance of the null hypothesis indicates that:

The slope of SLR model is zero / null

a. TRUE

Given: n = 25,

The 99% CI for mean:

where t_0.01,25-1 = t_0.01,24 = 2.797 (from t table)

Substituting the values:

We are given that the length of CI is 4:

i.e. [(2.5 + 0.559s) - (2.5 - 0.559s)] = 4

[2.5 + 0.559s - 2.5 + 0.559s] = 4

1.119s = 4

s = 4 / 1.03

= 3.58

The correct option would be c) s = 3.58 cm

3. The 99% Prediction interval can be computed using the formula:

Substituting the values,

= (-7.71, 12.71)

The range of the interval is obtained as:

12.71 - (-7.71)

= 12.71 +7.71

= 20.42

Hence the correct option would be

c) 20.42 cm