The time taken for Google to perform a Web crawl to update its search index is Normally distributed, with a mean of 7.2 days and standard deviation of 2 days.
i. What is the probability of Google’s web crawl taking less than 7 days?
ii. What is the probability of Google’s web crawl taking between 7 and 8 days?
iii.What is the first quartile of Google’s Web crawl time in days?
iv. If we sample 16 Web crawls taken by Google, what is the probability that the sample mean crawl time is less than 7.5 days?
You have a normal distribution with mean 7.2 and standard deviation of 2. If you use MS Excel the calculations would look like:
i. You take the cumulative normal distribution function with the standartized Z score as input. It gives you the left tail of the distribution up to 7.
=NORM.S.DIST((7-7.2)/2, TRUE) = 46%
ii. You calculate the left tail up to 8 as in i. and substract the left tail up to 7.
=NORM.S.DIST((8-7.2)/2,TRUE)-NORM.S.DIST((7-7.2)/2,TRUE) = 19.5%
iii. Use the inverse cumulative distribution function and scale up the result to your distribution (mean 7.2, std. deviation 2)
=NORM.S.INV(0.25)*2+7.2 = 5.85
iv. You have a sampling distribution. Its mean is equal to the mean of the original distribution (7.2). The standard error of the mean (this is what we call the standard deviation of the sampling distribution) is equal to the standard deviation of the original distribution over SQRT(number of samples/draws). Therefore:
=+NORM.S.DIST((7.5-7.2)/(2/SQRT(16)),TRUE) = 72.5%