Question

The data below were obtained to predict the sound frequency to which a person’s ear will respond based upon the length of time a person has been exposed to a high level of noise. Here “length of exposure” is the amount of time in weeks that a person has been living close to a major airport and the “hearing range” is reported in thousand cycles per second. Length of exposure In weeks (Left) Hearing range In thousand cycles per second (Right)

47 15.1

56 14.1

116 13.2

178 12.7

19 14.6

75 13.8

160 11.9

31 14.8
12 15.3

164 12.6

43 14.7

74 14.0

a. Find the correlation coefficient r.

b. Identify our x, explanatory variable and y, response variable and construct a scatter plot of y vs x.

c. Determine least squares equation that can be used for predicting a value of y based on a value of x.

d. Does this slope (in c) differ significantly from 0. Use alpha=0.05.

e. Use your model to predict y when x is i) 5, ii) 100. f. Provide 95% confidence intervals for the means (e, ii). g. Provide 95% prediction intervals for (e, ii).

h. Comment on your findings in (f) and (g). i. What were the basic assumptions for the model in (c)?

j. Would you consider the model in (b) to be a good model? Why or why not?

Answer 1

a) Correlation Coefficient ,r = -0.9476

Therefor we can conclude that the Length of exposure In weeks and Hearing range In thousand cycles per second is Negatively correlated.

b) Explanatory Variable, x = Length of exposure (Since we can find the hearing range only with the help of exposure)

   Response Variable,y = Hearing range In thousand cycles per second.

c)

Least Square Equation for predicting Y = 15.321834 - 0.017499 (x)

d)

The Slope(exp) significantly differs from 0 since the p value < 0.05 at 5% level of significance.

e)

Y = 15.321834 - 0.017499 (x = 5) = 15.234339

Y = 15.321834 - 0.017499 (x = 100) = 13.571934

f) 95% confidence interval for (x=100) = ( 13.32482 , 13.81895)

g) 95% prediction interval for (x=100) = ( 12.72298 , 14.42078)

h)

There is 95% confidence of the means of the model lies between ( 13.32482 , 13.81895) when the length of the exposure becomes 100.

There is 95% prediction of the means of the model lies between( 12.72298 , 14.42078) when the length of the exposure becomes 100.

i)

1. Linear relationship.
2. Multivariate normality.
3. No Multicollinearity.
4. No Auto-correlation.
5. Homoscedasticity.

j)

The model in (b) is not good because the variables are about to strongly negatively correlated.There seems to be no association between the length of exposure and hearing range in seconds.