Question

A few years ago, a number of prominent news sources published articles with provocative titles such as “You Now Have a Shorter Attention Span Than a Goldfish” (Time Magazine, 5/14/2015). These articles referenced a Microsoft study that claimed: • The average attention span of humans was 12 seconds in 2000. • The average attention span of humans was 8 seconds in 2013. • The average attention span of a goldfish is 9 seconds. Suppose all attention spans are normally distributed with a standard deviation of 2.1 seconds. (a) Assuming the numbers above are accurate, what is the probability a random person’s attention span in 2013 was greater than 9.5 seconds? (b) What is the probability that a randomly selected group of ten people in 2013 had an average attention span greater than 9.5 seconds? (c) A random group of 16 Denison students is found to have an average attention span of 10.3 seconds. Perform a hypothesis test to determine if this is statistically significant evidence at the α = 0.05 level to conclude Denison students have a longer attention span than a goldfish. (d) Are there any problems with the reported study? Do some internet searching to find problems with the figures presented.

Answer 1

Let X denote attention span of humans in 2000, and let Y denote attention span of humans on 2013

Given,

Part a

To Calculate,

                          ,                  where denotes the CDF of standard normal distribution.
                           ,          Calculated using Standard Normal tables.

Part b

We have a random sample of size 10, i.e. we have

Now,

i.e,

To compute,

               , Calculated using Standard Normal tables.

Part c

To test,

The test statistic is given as,

, which under the null hypothesis follows a Standard Normal Distribution.

Thus we have,

The p-value of the test is obtained as
Since the p-value is less than α = 0.05 , thus we can conclude that on the light of the given data, we do not have enough evidence to reject the null hypothesis. i.e, We cannot conclude that Denison students have a longer attention span than a goldfish.