How is grit related to an individual's overall personal achievement, which includes income, happiness, health, family and friendship? A government agency has conducted a survey on a simple random sample of individuals on grit and summarised an achievement index for them.

A least squares (LS) estimation is applied to estimate the regression model

E(AchievementiGriti)=0+1GritiE(AchievementiGriti)=0+1Griti

Some summary statistics are shown in the following table.

	Grit	Achievement
mean	65.300	35.461
var	419.426	69.006
cov	157.599
sum of squares	421097
sample size (n)	90
sum of squared residuals	871.166

Use the information to answer all the questions below.

(a) Fill the blanks in the regression table.

table	estimate	s.e.	t	p-value
intercept	[a]	[b]	[c]	[d]
slope	[e]	[f]	[g]	[h]

In your answer, you may arrange the values as in the above output table or simply write [a]=XXX, ..., [h]=XXX.

(b) What is the value of R2R2

(c) What is the underlying null hypothesis for 11 from the t-statistic in the Table of question (a)?

(d) What is the underlying alternative hypothesis for 11from the p-values in the Table of question (a)?

(e) Compute the 99% confidence interval for 11 and explain its interpretation. Show the steps of calculation.

(f) Carry out a test to investigate if high Grit is associated with high achievement at 5% significance level. You need to

(1) define new notation (if any), (2) briefly justify your testing procedure, (3) state the hypotheses, (4) calculate the test statistic and show your calculation steps, (5) use the p-value approach, (6) make decision for the test and briefly explain.