Question

The horsepower (Y, in bhp) of a motor car engine was measured at a chosen set of values of running speed (X, in rpm). The data are given below (the first row is the running speed in rpm and the second row is the horsepower in bhp):

rpm	1100	1400	1700	2300	2700	3200	3500	4000	4300	4600	4900	5200	6100
Horsepower (bhp)	41.11	50.41	89.98	97.76	143.29	168.77	171.62	199.89	207.23	233.34	241.96	252.73	311.44

The mean and sum of squares of the rpm are 3461.53853461.5385 rpm and 185040000.0000185040000.0000 rpm 22 respectively; the mean of the horsepower values is 169.9638169.9638 bhp and the sum of the products of the two variables is 9184823.00009184823.0000 rpm bhp. A scatterplot displaying the data is shown below:



Please provide answers to the following to 3 decimals places where appropriate:

Part a)

Compute the regression line for these data, and provide your estimates of the slope and intercept parameters. To calculate the slope and intercept please keep intermediate results to 6 decimal places (when you calculate the intercept do not round the value of the slope).

Slope:

Intercept:

Note: For the parts below, use the slope and intercept values in Part a, corrected to 3 decimal places to calculate answers.

Part b)

Based on the regression model, what level of horsepower would you expect the engine to produce if running at 24002400 rpm?

Answer:

Part c)

Assuming the model you have fitted, if increase the running speed by 100100 rpm, what would you expect the change in horsepower to be?

Answer:

Part d)

The standard error of the estimate of the slope coefficient was found to be 0.0016930.001693. Provide a 95% confidence interval for the true underlying slope.

Confidence interval: (  ,  )

Part e)

Without extending beyond the existing range of speed values or changing the number of observations, we would expect that increasing the variance of the rpm speeds at which the horsepower levels were found would make the confidence interval in (d)

A. narrower.
B. either wider or narrower depending on the values chosen.
C. unchanged.
D. wider.


Part f)

If testing the null hypothesis that horsepower does not depend linearly on rpm, what would be your test statistic? (For this part, you are to calculate the test statistic by hand using appropriate values from the answers you provided in part (a) accurate to 3 decimal places, and values given to you in part (d).)

Answer:

Part g)

Assuming the test is at the 1% significance level, what would you conclude from the above hypothesis test?

A. Since the observed test statistic does not fall in either the upper or lower 1 percentiles of the t distribution with 1111 degrees of freedom, we cannot reject the null hypothesis that the horsepower does not depend linearly on rpm.
B. Since the observed test statistic does not fall in either the upper or lower 1/2 percentiles of the t distribution with 1111 degrees of freedom, we can reject the null hypothesis that the horsepower does not depend linearly on rpm.
C. Since the observed test statistic falls in either the upper or lower 1/2 percentiles of the t distribution with 1111 degrees of freedom, we cannot reject the null hypothesis that the horsepower does not depend linearly on rpm.
D. Since the observed test statistic falls in either the upper or lower 1/2 percentiles of the t distribution with 1111 degrees of freedom, we can reject the null hypothesis that the horsepower does not depend linearly on rpm.
E. Since the observed test statistic does not fall in either the upper or lower 1/2 percentiles of the t distribution with 1111 degrees of freedom, we cannot reject the null hypothesis that the horsepower does not depend linearly on rpm.

Answer 1

t table as follows: