Dataset: Highway_Speeds
A police officer is concerned about speeds on the Newburgh-New Paltz section of NYS Thruway I-87. The data accompanying this exercise show the speeds of 40 cars on a Saturday afternoon. The speed limit on this portion of I-87 is 65 mph. Test if the average speed is greater than the speed limit, at \(\alpha = 0.01\). Are the officer’s concerns warranted?
We will use the second column, and rename the column to “Speed”:
colnames(Highway_Speeds)[2]="Speed"
head(Highway_Speeds)## Highway.1..55.mph. Speed
## 1 60 70
## 2 55 65
## 3 53 65
## 4 65 62
## 5 57 70
## 6 58 64Step 1
x_bar = mean(Highway_Speeds$Speed) = 66
s = sd(Highway_Speeds$Speed) = 3.00427046479524
n = 40
mu_0 = 65
df = n-1 = 39
\(\alpha = 0.01\).
Step 2: Hypothesis
\(H_0:\mu \le 65\); \(H_1:\mu > 65\)
Step 3: Standard Error
SE = s/sqrt(n)
SE = 0.475016868796284
Step 4: T-statistics
Tstat = (x_bar - mu_0)/SE
Tstat = 2.10518839580171
Step 5: find pvalue
pvalue = pt(Tstat,df,lower.tail=FALSE)
pvalue = 0.0208836093938327
Conclusion:
Since p_value (0.0208836) > \(\alpha\) (0.01), do not reject the null hypothesis. At the 1% significance level, we can conclude that the average speed of cars is 65 mph or lower, so officers should not be concerned.
if(!require(BSDA)) install.packages(“BSDA”) library(BSDA)
Step 1: Hypothesis
\(H_0:\mu \le 65\); \(H_1:\mu > 65\)
Step 2: Find sample size and alpha
n= nrow(Highway_Speeds)
n = 40
\(\alpha = 0.01\)
Step 3: Use t-test
results = t.test(Highway_Speeds$Speed,alternative=“greater”,mu=65,conf.level=0.99)
Tstat = results$statistic
pvalue = results$p.value
Conclusion: Since p_value (0.0208836) > \(\alpha\) (0.01), do not reject the null hypothesis. At the 1% significance level, we can conclude that the average speed of cars is 65 mph or lower, so officers should not be concerned.