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1. The following information is obtained for a sample of 16 observations taken from a population
; 
; and 
a) Make a 99% confidence interval for 
.
b) Using a significance level of 0.025, test the null hypothesis that is zero against the alternative that is positive.
c) Using a significance level of 0.01, can you conclude that is zero against the alternative that it is different from zero?
d) Using a significance level of 0.02, test whether is different from 4.50.
2. The following information is obtained for a sample of 100 observations taken from a population. (Note that because 
, we can use the normal distribution to make a confidence interval and test a hypothesis about )
; 
; and 
a) Make a 98% confidence interval for
b) Test at the 2% significance level whether is zero against the alternative that it is positive.
c) Can you conclude that is zero? Use 
.
d) Using a significance level of 0.01, test whether is 1.75 against the alternative that it is greater than 1.75.
3. Refer to exercise 7 of previous chapter. The following table which gives the ages (in years) and prices (in hundred of dollars) of eight cars of specific model, is reproduced from that exercise.
| Age | ||||||||
| Price |
a) Construct a 95% confidence interval for .
b) Test at the 5% significance level if is zero against the alternative that it is negative.
4. The following table gives the experience (in years) and monthly salaries (in thousands of tenge) of nine randomly selected secretaries
| Experience | |||||||||
| Monthly salary |
a) Find the least squares regression line with experience as an independent and monthly salary as dependent variables.
b) Construct a 95% confidence interval for .
c) Test at the 2.5% significance level if is zero against the alternative that it is positive.
5. The data on the size of six offices (in hundreds of square meters) and the monthly rents (in dollars) paid by firms for those offices are reproduced below from exercise 6 of the previous section.
| Size of offices | ||||||
| Monthly rent |
a) Construct a 99% confidence interval for . You can use the calculations made in exercise 6 of previous section here.
b) Test at the 5% significance level the null hypothesis that is zero against the alternative that it is different from zero.
6. The following data give information on the ages (in years) and the number of breakdowns during the past year for a sample of six machines at a large company.
| Age | ||||||
| Number of breakdowns |
a) Find the least squares regression line 
b) Give a brief interpretation of the values a and b.
c) Compute and interpret 
.
d) Compute the standard deviation of errors.
e) Construct a 98% confidence interval for .
f) Test at the 2.5% significance level the null hypothesis that is zero against the alternative that it is positive.
7. The following table gives information on the temperature in a city and volume of the ice cream (in thousands) sold at the supermarket for a random sample of eight days during the summer.
| Temperature | ||||||||
| Ice cream sold | 3.64 | 3.12 | 4.08 | 2.84 | 3.98 | 3.55 | 4.02 | 4.38 |
a) Find the least squares regression line . Take temperature as an independent variable and volume of ice cream sold as a dependent variable.
b) Give a brief interpretation of the values a and b.
c) Compute and interpret .
d) Compute the standard deviation of errors.
e) Construct a 95% confidence interval for .
f) Test at the 1% significance level the null hypothesis that is zero against the alternative that it is positive.
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| Statistical inference: Hypothesis tests and confidence intervals | | | Using the regression model for prediction a particular value of y |