5.6 Working backwards, Part II. A 90% confidence interval for a population mean is (65, 77). The population distribution is approximately normal and the population standard deviation is unknown. This confidence interval is based on a simple random sample of 25 observations. Calculate the sample mean, the margin of error, and the sample standard deviation.
90% CI is (65, 77) SD is unknown 25 observations
sample mean is (x2 + x1)/2
n <- 25
x1 <- 65
x2 <- 77
(x2 + x1)/2## [1] 71The sample mean is 71
Margin of Error is (x2 - x1)/2
n <- 25
x1 <- 65
x2 <- 77
ME <- (x2 - x1)/2The margin of error is 6
Sample SD ME = t*
df <- 25 - 1
p <- 0.9
p2tails <- p + (1-p)/2
tval <- qt(p2tails, df)
SE <- ME / tval
sd <- SE * sqrt(n)
sd## [1] 17.534815.14 SAT scores. SAT scores of students at an Ivy League college are distributed with a standard deviation of 250 points. Two statistics students, Raina and Luke, want to estimate the average SAT score of students at this college as part of a class project. They want their margin of error to be no more than 25 points.
- Raina wants to use a 90% confidence interval. How large a sample should she collect?
z <- 1.645
sd <- 250
ME <- 25
n <- ((z * sd) / ME ) ^ 2
n## [1] 270.6025- Luke wants to use a 99% confidence interval. Without calculating the actual sample size, determine whether his sample should be larger or smaller than Raina’s, and explain your reasoning.
His sample should be larger since it will require a higher z number.
- Calculate the minimum required sample size for Luke
z <- 2.575
sd <- 250
ME <- 25
n <- ((z * sd) / ME ) ^ 2
n## [1] 663.06255.20 High School and Beyond, Part I. The National Center of Education Statistics conducted a survey of high school seniors, collecting test data on reading, writing, and several other subjects. Here we examine a simple random sample of 200 students from this survey. Side-by-side box plots of reading and writing scores as well as a histogram of the differences in scores are shown below.
(a) Is there a clear difference in the average reading and writing scores?
There does not seem to be a clear difference between the scores.
(b) Are the reading and writing scores of each student independent of each other?
The scores are not independent of each other
(c) Create hypotheses appropriate for the following research question: is there an evident difference in the average scores of students in the reading and writing exam?
H0: There is no difference between the average reading and writing score.
HA: There is a difference between the average reading and writingscore.
(d) Check the conditions required to complete this test. Independence and normal distribution. The data is of a pair set, so not indepdent. THe data is reasonably normal.
(e) The average observed difference in scores is ¯xread???write = ???0.545, and the standard deviation of the differences is 8.887 points. Do these data provide convincing evidence of a difference between the average scores on the two exams?
sd <- 8.887
m <- -0.545
n <- 200
df <- n-1
SE <- sd / sqrt(n)
t <- m - 0/ SE
p <- 2* pt(t, df)The p-value is not less than 0.05, so there is not enough evidence of a difference in students reading and writing scores.
(f) What type of error might we have made? Explain what the error means in the context of the application.
Type I error: Incorrectly reject the null hypothesis.
Type II error: Incorrectly reject the alternative hypothesis.
Could have possibly made a type II error by rejecting the alternative hypothesis HA.
(g) Based on the results of this hypothesis test, would you expect a confidence interval for the average difference between the reading and writing scores to include 0? Explain your reasoning
Yes, I would expect a confidence interval for the average difference between reading and writing scores to include 0.
5.32 Fuel efficiency of manual and automatic cars, Part I. Each year the US Environmental Protection Agency (EPA) releases fuel economy data on cars manufactured in that year. Below are summary statistics on fuel efficiency (in miles/gallon) from random samples of cars with manual and automatic transmissions manufactured in 2012. 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? Assume that conditions for inference are satisfied.
n <- 26
mean_a <- 16.12
sd_a <- 3.58
mean_m <- 19.85
sd_m <- 4.51
#difference
mu_diff <- mean_a - mean_m
#standard error
SE <- ((sd_a ^ 2 / n) + (sd_m ^ 2/n) )^0.5
t <- (mu_diff - 0) / SE
df <- n - 1
p <- pt(t, df = df)
p## [1] 0.001441807Since the p-value is less than 0.05, we reject the null hypothesis.
5.48. The General Social Survey collects data on demographics, education, and work, among many other characteristics of US residents.47 Using ANOVA, we can consider educational attainment levels for all 1,172 respondents at once. Below are the distributions of hours worked by educational attainment and relevant summary statistics that will be helpful in carrying out this analysis.
- Write hypotheses for evaluating whether the average number of hours worked varies across the five groups.
H0: The difference of all averages is equal Ha: ONe average is not equal to the others
- Check conditions and describe any assumptions you must make to proceed with the test.
Data within each group is nearly normal.
The observations within the data is independent.
Below is part of the output associated with this test. Fill in the empty cells.
df Sum Sq MeanSq F Value Pr(>f)
degree 4 2006.16 501.54 2.188984 0.0682
res 1167 267382 229.12
total 1171 269388.16
- What is the conclusion of the test? The p-value is greater than 0.05, we can conclude that there is not a significant difference between the groups. Null hyp not rejected.