Question

Use R studio to answer the following questions

location 1 location 2 63.9 66.8 71.3 63.8 70.6 60.7 54.6 66.2 63.7 59.5 61 59.5 69.8 66.1 72.4 64.6 64.4 64.8 69.8 61.6 65.6 61.3 62.8 64.9 63.6 66.2

Temperatures are extremely variable in sections of the Colorado River near the Glen Canyon Dam.  Researchers measure temperature in two locations (1 and 2) with samples of size n₁ = 10 and n₂ = 16, and calculate variances of s₁² = 20 °C ²and s₂² = 15 °C ².

1. Construct a 98% confidence interval for the ratio s₁²/s₂². (Hint: copy this data into a .csv file and read into R, or read in as vectors.)
2. Interpret this interval.
3. Test the hypothesis that the two variances are equal at a = 0.10. Be sure to list (i) your hypotheses, (ii) test statistic, (iii) decision, and (iv) conclusion.

Answer 1

using R we have

> location1=c(63.9,63.8,54.6,59.5,69.8,64.6,69.8,65.6,62.8,63.6)
> location2=c(66.8,70.6,66.2,61,66.1,64.4,61.6,61.3,64.9,66.2,71.3,60.7,63.7,59.5,72.4,64.8)
> var.test(location1, location2, ratio = 1, alternative = c("two.sided"),conf.level = 0.98)

   F test to compare two variances

data: location1 and location2
F = 1.3332, num df = 9, denom df = 15, p-value = 0.598
alternative hypothesis: true ratio of variances is not equal to 1
98 percent confidence interval:
0.342316 6.615682
sample estimates:
ratio of variances
1.333248

1) a 98% confidence interval for the ratio s12/s22 is  (0.342316 , 6.615682)

2 ) we are 98% confident that the the ratio of population variances of two locations lies in between  (0.342316 , 6.615682) .

3)the null and alternative hypotheses is

Ho:

Ha:

(ii) test statistic from R is 1.3332

p value is 0.598

(iii) decision: since p valjue is greater than 0.10 so we don not reject Ho

(iv) conclusion: we wil conclude that the population variances of two locations are equal