Question

In 2011, 51% of cell phone owners in a country reported that their cell phone was a smartphone. The following year, the researchers wanted to test to see if the proportion of cell phone owners in that country who have a smartphone has increased over time at the a = 0.05 level. They surveyed a random sample of 934 cell phone owners in that country and found that 501 of them had a smartphone. They conducted a significance test and found the p-value to be 0.0532. Is there convincing evidence that the proportion of cell phone owners in that country who have a smartphone has increased over time?

Answer 1

Answer:

The calculated value Z = 1.5950 < 1.96 at 0.05 level of significance

The null hypothesis is accepted at a 0.05 level of significance

The proportion of cell phone owners in that country who have a smartphone has not increased over time

Step-by-step explanation:

Step(i):-

Given that the random sample size 'n' = 934

Given that the population proportion P = 0.51

                                                            Q= 1-P

                                                           Q = 1- 0.51 = 0.49

The sample proportion

                                   $p = \frac{x}{n} = \frac{501}{934} = 0.5364$

Level of significance = 0.05

Critical value Z₀.₀₅ = 1.96

Step(ii):-

Null hypothesis:  P < 0.51

Alternative Hypothesis : P > 0.51

  Test statistic

                    $Z = \frac{p-P}{\sqrt{\frac{PQ}{n} } }$

                   $Z = \frac{0.536-0.51}{\sqrt{\frac{0.51 x 0.49}{934} } }$

                  Z =  1.5950

Final answer:-

Given that P-value 0. 0532

P-value > 0.05

Rejected Alternative Hypothesis and accepted null hypothesis

               ( OR)

The calculated value Z = 1.5950 < 1.96 at 0.05 level of significance

The null hypothesis is accepted at a 0.05 level of significance

The proportion of cell phone owners in that country who have a smartphone has not increased over time