5.6, 5.14, 5.20, 5.32, 5.48

5.6: Working backwards, Part II

Sample mean:
\[ \frac{(65 + 77)}{2} = 71 \]

Margin of error:
\[ 77 - 71 = 6 \]

Sample standard deviation:
\[ \frac{6}{2.06} * \sqrt(25) = 14.56 \]

5.14: SAT scores

  1. Raina should collect a sample size of 271 students.

  2. Luke will need to collect a larger sample size than Raina, because he has a higher confidence level. With a larger confidence level, we should collect a larger sample size to be absolutely sure we’re capturing the population mean within our parameters.

  3. Luke should collect a sample size of 663 students.

5.20: High School and Beyond, Part I

  1. There is a slight difference in the average reading and writing scores, but it’s not enormous.

  2. The reading and writing scores of each student are independent of each other.

  3. \[ H_o: reading = writing \\ H_a: reading \neq writing \]

  4. Conditions: Sample observations are independent The reading and writing scores of each student are independent of each other. Sample size is large (n >= 30)
    Our sample size is 200, so this condition is met. Population distribution is not strongly skewed.
    The differences in scores are normally distributed.

\[ \frac{(0.545 - 0)}{8.887/\sqrt(200)} = 0.87 \]

The data do not provide convincing evidence of a difference between the average scores on the two exams.

  1. We may have made a type II error. This means we have not rejected the null hypothesis when indeed we should have.

  2. Based on the results of this hypothesis test, I would expect a confidence interval for the average difference between the reading and writing scores to include 0.

5.32: Fuel efficiency of manual and automatic cars, Part I

Do these data provide strong evidence of a difference between the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage?

\[ H_o: manual = automatic \\ H_a: manual \neq automatic \]

Given our calculated T-value of 3.3 and our 25 degrees of freedom, the data provide enough evidence to reject the null hypothesis. There is a statistically significant difference in the gas mileage of manual and automatic vehicles.

5.48: Work hours and education

  1. \[ H_o: \mu_1 = \mu_2 = \mu_3 = \mu_4 = \mu_5 \\ H_a: \mu_1 \neq \mu_2 \neq \mu_3 \neq \mu_4 \neq \mu_5 \]

  2. Conditions: Sample observations are independent The samples are independent from each other. Sample size is large (n >= 30)
    Our sample size is 1,172, so this condition is met. Population distribution is not strongly skewed.
    The population is fairly normal. Even if it wasn’t, we have enough samples to handle a skew.

  3.             | Df    | Sum Sq  | Mean Sq | F value | Pr(>F)
----------------------------------------------------------
degree      | 4     | 2004.1  | 501.03  | 2.19    | 0.0682
Residuals   | 1167  | 267382  | 229.12  |
----------------------------------------------------------
Total       | 1171  | 269386.1

  4. At 95% confidence, the data do not provide enough evidence to reject the null hypothesis.

z <- qnorm(.95, mean = 0, sd = 1)
(250/(25/z))^2
## [1] 270.5543
z <- qnorm(.995, mean = 0, sd = 1)
(250/(25/z))^2
## [1] 663.4897
8.887/sqrt(200)
## [1] 0.6284058
(0.545 - 0)/(8.887/sqrt(200))
## [1] 0.867274
sqrt((3.58^2/26) + (4.51^2/26))
## [1] 1.12927
(19.85 - 16.12 - 0)/(sqrt((3.58^2/26) + (4.51^2/26)))
## [1] 3.30302