Question

A telecom research group stated that 30% of households have three cell phones. A cell phone
company has reason to believe that the proportion is significantly different from 30%. To assess
this claim, they conduct a hypothesis test. Their marketing department surveys 175 households
with the result that 63 of the households have three cell phones. Using a significance level of
0.05, determine the reasonableness of the cell phone company’s claim.





3. Sports fans are led to believe the average playing career of a running back is 4 years, however
the players association believes the actual playing career to be significantly less. To test this
belief, a random sample of 36 retired players were surveyed, and the average playing career was
3.74 years with a sample standard deviation of 1.21 years. Use the Hypothesis Testing
procedure with a 2% level of significance to determine whether the playing career is 4 years and
state a conclusion.

Answer 1

$1) H_0: p=0.3, H_1: p\neq 0.3 \text{ (2-tailed test)}$

$pN=(0.3)175=5.25>5\\ (1-p)N=(0.7)175>5$

So we will use z-test.

$z-value=1.96$ ($\alpha=0.05)$

If test statistic $>1.96$ or $<-1.96$ we shall reject $H_0.$ Otherwise we accept $H_0.$

$z=\frac{p^*-p}{\sqrt{\frac{p(1-p)}{N}}}=\frac{\frac{63}{175}-0.3}{\sqrt{\frac{(0.3)(0.7)}{175}}}\approx 1.732\text{ --- test statistic}.$

So we accept $H_0.$

The claim of the cell phone company is not reasonable.

2) $H_0: a=a_0=4, H_1: a<a_0=4.$

We shall use a random variable $T=\frac{(\overline{x}-a_0)\sqrt{n}}{s}$ and t-test;

$k=n-1=36-1=35\text{ --- degree of freedom}.$

$t_{cr}=t_{cr}(\alpha;k)=t_{cr}(0.02;35)\approx 2.1332\text{ (one-sided)}.\\ t_{obs}=\frac{(3.74-4)\sqrt{36}}{1.21}\approx -1.289.\\ t_{obs}=-1.289>-t_{cr}=-2.1332.$

So $t_{obs}\text{ is not in the critical region } (-\infty,-2.1332)\text{ and we accept } H_0.$

Association's claim is not reasonable.