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(Achievementi∣Griti)=β0+β1Griti Some summary statistics are shown in the following table. Grit Achievement mean 64.86 34.87 var 456.3 70.10 cov 165.76 sum of squares 409942 sample size (n) 88 sum of squared residuals 859.65 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 R2 (c) What is the underlying null hypothesis for β1 from the t-statistic in the Table of question (a)? (d) What is the underlying alternative hypothesis for β1from the p-values in the Table of question (a)? (e) Compute the 90% confidence interval for β1and explain its interpretation. Show the steps of calculation. (f) Carry out a test to investigate if high Grit is associated with high achievement at 1% 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.
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(Achievementi∣Griti)=β0+β1Griti
Some summary statistics are shown in the following table.
| Grit | Achievement | |
| mean | 64.86 | 34.87 |
| var | 456.3 | 70.10 |
| cov | 165.76 | |
| sum of squares | 409942 | |
| 88 | ||
| sum of squared residuals | 859.65 |
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 R2
(c) What is the underlying null hypothesis for β1 from the t-statistic in the Table of question (a)?
(d) What is the underlying alternative hypothesis for β1from the p-values in the Table of question (a)?
(e) Compute the 90% confidence interval for β1and explain its interpretation. Show the steps of calculation.
(f) Carry out a test to investigate if high Grit is associated with high achievement at 1% 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.
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