Question

Registrar’s Office analyzed a sample of n = 20 students. They set control variable X = number of sessions missed and recorded response variable Y = final exam score. Linear regression model produced the fitted line equation

Yˆ = 90 − 1.2 · X with SE = SE [b1] = 0.4. 1.

1.Estimate the population slope for this model with confidence = C = 0.95. Derive upper and lower limits of the interval.

2. At significance level α = 0.05, do you have enough evidence that X negatively influences on Y ? Show critical value, test statistic, and state rejection rule to justify your decision.

Answer 1

Given -> Registrar’s Office analysed a sample of n = 20 students.

The control variable X = number of sessions missed. Response variable Y = final exam score.

Linear regression model produced the fitted line equation

Yˆ = 90 − 1.2 * X

Standard error = SE [β1] = 0.4

1) From the regression equation provide we can observe estimated population slope β1 = -1.2

Now let’s derive upper and lower limits of the confidence interval with confidence = C = 0.95

Therefore α = 0.05

The 95% confidence interval for β1 = β1 + Margin of error

Margin of error = critical value * standard error

Standard error = 0.4

Critical Value at α = 0.05 and degrees of freedom (df) = n-2 = 20-2 =18

Critical value = 2.100922

Margin of error = 2.100922* 0.4 =0.8403688

The 95% confidence interval for β1 = β1 + Margin of error

= -1.2 + 0.8403688

Lower limit of confidence interval = -1.2 - 0.8403688 = -2.0403688

Upper limit of confidence interval = -1.2 + 0.8403688 = -0.3596312

95% confidence interval: 2.04037 ≤ β1 ≤ -0.35963

2) At significance level α = 0.05, we need to test significance of β1.

H0: The slope of the regression line is equal to zero. (β1 = 0)

H1: The slope of the regression line is less than zero. ((β1 < 0)

Test statistic = t = β1/SE

= -1.2 / 0.4

t= -3

Critical Value at α = 0.05 and degrees of freedom (df) =18

t Critical = 2.100922

The p-value = p(t < 2.100922 ) = 0.003843

The t ( -3) < t Critical ( 2.100922) and p value (0.003843) is < α = 0.05 we reject H0.

There significant statistical evidence that support the claim that is X negatively influences on Y.