Question

sted below are speeds​ (mi/h) measured from traffic on a busy highway. This simple random sample was obtained at​ 3:30 P.M. on a weekday. Use the sample data to construct

aa

98​%

confidence interval estimate of the population standard deviation.

6161

6464

6464

5757

6464

5454

6060

5959

6060

7070

6262

6868

Question

The confidence interval estimate is

​mi/h ___<σ< ​___mi/h.

​(Round to one decimal place as​ needed.)

Does the confidence interval describe the standard deviation for all times during the​ week? Choose the correct answer below.

A.

Yes. The confidence interval describes the standard deviation for all times during the week.

B.

No. The confidence interval is an estimate of the standard deviation of the population of speeds at​ 3:30 on a​ weekday, not other times.

Answer 1

Confidence Interval Formula for σ is as follows:

Square Root((n - 1)s^2/χ^2α/2) < σ < Square Root((n-1)s^2/χ^21 - α/2)

where: (n - 1) = Degrees of Freedom, s^2 = sample variance and α = 1 - Confidence Percentage

STEP1: First find degrees of freedom:

Degrees of Freedom = n - 1

Degrees of Freedom = 12 - 1

Degrees of Freedom = 11

STEP 2: Calculate α:

α = 1 - confidence%

α = 1 - 0.98

α = 0.02

STEP 3: Find low end confidence interval value:

αlow end = α/2

αlow end = 0.02/2

αlow end = 0.01

Find low end χ2 value for 0.01 χ^2(0.01) = 24.725 <--- Value can be found on Excel using =CHIINV(0.01,11)

STEP 5: Calculate low end confidence interval total

: Low End = Square Root((n - 1)s^2/χ^2(α/2))

Low End = √(11)(19.90)/24.725)

Low End = √218.9/24.725

Low End = √8.8533872598584

Low End = 2.9755

STEP 6: Find high end confidence interval value:

αhigh end = 1 - α/2

αhigh end = 1 - 0.02/2

αhigh end = 0.99

Find high end χ2 value for 0.99 χ^2(0.99) = 3.0535 <--- Value can be found on Excel using =CHIINV(0.99,11)

STEP7: Calculate high end confidence interval total:

High End = Square Root((n - 1)s^2/χ^2(1 - α/2))

High End = √(11)(19.90)/3.0535)

High End = √218.9/3.0535

High End = √71.688226625184

High End = 8.4669

Now our interval 3.0 < σ < 8.5 <---- This is our 98% confidence interval

No. The confidence interval is an estimate of the standard deviation of the population of speeds at​ 3:30 on a​ weekday, not other times.