MATH SOLVE

4 months ago

Q:
# A recent broadcast of a television show had a 10 share, meaning that among 6000 monitored households with TV sets in use, 10% of them were tuned to this program. Use a 0.01 significance level to test the claim of an advertiser that among the households with TV sets in use, less than 25% were tuned into the program. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the P-value method. Use the normal distribution as an approximation of the binomial distribution.

Accepted Solution

A:

Answer:Null and alternative hypothesis[tex]H_0: \pi \geq0.25\\\\H_1: \pi<0.25[/tex]Test statistic [tex]z=-26.82[/tex]P-value [tex]P=0[/tex]The null hypothesis is rejected.It is concluded that the proportion of households tuned into the program is less than 25%. The claim of the advertiser is rigth and got statistical support.Step-by-step explanation:We have to perform a hypothesis test of a proportion.The claim is that less than 25% were tuned into the program, so we will state this null and alternative hypothesis:[tex]H_0: \pi \geq0.25\\\\H_1: \pi<0.25[/tex]The signifiance level is 0.01.The sample has a proportion p=0.1 and sample size of n=6000.The standard deviation of the proportion, needed to calculate the test statistic, is:[tex]\sigma_p=\sqrt{\frac{\pi(1-\pi)}{N} } =\sqrt{\frac{0.25(1-25)}{6000} } =0.0056[/tex]The test statistic is calculated as:[tex]z=\frac{p-\pi+0.5/N}{\sigma_p} =\frac{0.1-0.25+0.5/6000}{0.0056}=\frac{-0.1499}{0.0056} =-26.82[/tex]As this is a one-tailed test, the P-value is P(z<-26.82)=0. The P-value is smaller than the significance level (0.01), so the effect is significant.Since the effect is significant, the null hypothesis is rejected. It is concluded that the proportion of households tuned into the program is less than 25%. The claim of the advertiser is rigth and got statistical support.