Question

Internet company Gurgle is carrying out testing on the efficiency of its search engine. A sample of 31 searches have been carried out and the time taken to display the results has been recorded for each search. The mean search time for the sample was calculated as 0.2258 seconds. The standard deviation of the search times for the sample was calculated as 0.0188 seconds. The population standard deviation of search times is unknown.
1. Select all the techniques that are commonly used to construct a confidence interval for the mean when the population standard deviation (σ) is unknown:

a. Approximate the population standard deviation (σ) with the sample standard deviation (s)
b. Replace the sample size (n) with n-1
c. Decrease the confidence level to compensate for the increased margin of error
d. Approximate the standard normal distribution with the Student's t distribution

2. Calculate the upper and lower bounds of the 95% confidence interval for the mean search time for the Gurgle search engine.

Answer 1

Answer:

d. Approximate the standard normal distribution with the Student's t distribution

(0.2199 ; 0.2327)

Step-by-step explanation:

Given that :

Sample size, n = 31

Sample mean, xbar = 0.2258

Sample standard deviation, s = 0.0188

Confidence interval (C. I) :

xbar ± margin of error

Margin of Error : Tcritical * s/sqrt(n)

Degree of freedom, df = n - 1 = 31 - 1 = 30

Tcritical value :

T0.05/2, 30 = 2.042

Margin of Error = 2.042 * 0.0188/sqrt(31)

Margin of Error = 0.0068949

C. I = 0.2258 ± 0.0068949

Lower boundary : (0.2258 - 0.006895) = 0.2189

Upper boundary : (0.2258 - 0.006895) = 0.2327

(0.2199 ; 0.2327)