Question

The population of adults with tablets and/or smartphones get an average of 6 hours (µ = 6 and σ = 2) of sleep a night. To see whether electronic use was interfering with sleep, researchers had a sample of 25 adults turn off their smartphones and tablets 1 hour before going to bed. This sample got an average of 6.9 hours of sleep a night. Assuming the population is normally distributed, use an alpha of .05 to test the hypothesis that turning off their electronics early increased the number of hours of sleep they got.

A.Specify the null and alternative hypotheses

B. Report the critical value, the SE, and the test statistic

C.Report your decision in a sentence

D. What type of error did you risk making?

Answer 1

Solution:

A. Specify the null and alternative hypotheses

Here, we have to use one sample z test for the population mean.

Null hypothesis: H₀: Turning off tablets and/or Smartphone’s early do not increase the number of hours of sleep.

Alternative hypothesis: H_a: Turning off tablets and/or Smartphone’s early increased the number of hours of sleep.

H₀: µ = 6 versus H_a: µ > 6

This is an upper tailed (one tailed) test.

B. Report the critical value, the SE, and the test statistic

We are given

Level of significance = α = 0.05

So, critical value by using z-table is given as below:

Critical Z value = 1.6449

SE = σ/sqrt(n)

We are given σ = 2, n = 25

SE = 2/sqrt(25) = 0.4000

Test statistic = Z = (Xbar - µ) / [σ/sqrt(n)]

Test statistic = (6.9 – 6)/[2/sqrt(25)]

Test statistic = 0.9/0.4

Test statistic = 2.25

C. Report your decision in a sentence

From above test statistic, we have

P-value = 0.0122

(by using z-table)

P-value < α = 0.05

Or

Test statistic = 2.25 > Critical Z value = 1.6449

So, we reject the null hypothesis

There is sufficient evidence to conclude that turning off tablets and/or Smartphone’s early increased the number of hours of sleep.

D. What type of error did you risk making?

We are making risk of type I error for the above test, because we reject the null hypothesis for above test.