Question #347108

A study believes that 70% of adults in the Philippines own a cellphone.

A cellphone manufacturer believes that the actual number is much

less than 70%. 100 Filipino adults were surveyed, of which 74 have

cellphones. Using a 5% level of significance, is the cellphone

manufacturer’s claim valid or not?


1
Expert's answer
2022-06-02T12:42:34-0400

Claim: the actual number is much less than 70%.

The following null and alternative hypotheses for the population proportion needs to be tested:

H0:p0.70H_0:p\ge0.70

Ha:p<0.70H_a:p<0.70

This corresponds to a left-tailed test, for which a z-test for one population proportion will be used.

Evidence:

Based on the information provided, the significance level is α=0.05,\alpha = 0.05 , and the critical value for a left-tailed test is zc=1.6449.z_c = -1.6449.

The rejection region for this left-tailed test is R={z:z<1.6449}.R = \{z: z < -1.6449\}.

The z-statistic is computed as follows:


z=p^p0p0(1p0)n=74/1000.70.7(10.7)100=2.1822z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{74/100-0.7}{\sqrt{\dfrac{0.7(1-0.7)}{100}}}=2.1822

Since it is observed that z=2.18221.6449=zc,z = 2.1822 \ge-1.6449= z_c, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is p=P(Z<2.1822)=0.985453,p=P(Z<2.1822)= 0.985453, and since p=0.985453>0.05=α,p= 0.985453>0.05=\alpha, it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population proportion pp is less than 0.70, at the α=0.05\alpha = 0.05 significance level.


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