Question

A recent sports game set a record for the number of television viewers. The game had a share of 78​%, meaning that among the television sets in use at the time of the​ game, 78​% were tuned to the game. The sample size is 25,563 households. Use a 0.01 significance level to test the claim that more than 74​%of television sets in use were tuned to the sports game. 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 and the normal distribution as an approximation to the binomial distribution.

1. The test statistic is z=____

2. The P-value=_____

Answer 1

Let x be the no. of viewers (people who were tuned to the game)

then x follows Binomial Distribution , with

mean = np = 25,563* 0.78 = 19939.14

when, p = 0.78 , then q = 1-0.78= 0.22

standard deviation = sqrt(npq) =

= 66.23

Now using the central limit theorem, the average no of people who were tuned to the game follows a normal distribution, because the sample size n =25563, which is a huge number

= the mean no. of people who were tuned

Here, mean of = np = 25563*0.78 = 19939.14

sd of = sd of x/ = 66.23/ = 66.23/159.88 = 0.4142

follows a normal distribution with mean of 19939.14 and sd of 0.4142

Now, we could do our hypothesis testing ,

since we now know the sample standard deviation, we would do a One sample t test

Steps:

1. Define Null and Alternative hypothesis

H0 : u= 0.74

HA : u 0.74

2. State alpha = 0.01, t critical = 1.96

3. Degrees of freedom = n-1 = 25,563-1

4. Decision Rule, reject the null if the t statistic is less than the t critical value

5. Test Statistic =

= 0.78-0.74/ 0.4142

=0.09657

0.096<1.96

6. Conclusion, failed to reject the null hypothesis