A company that produces laundry detergent is interested in the level of consumer awareness concerning the unique properties of their product. Historical data indicated that among the consumers who buy this detergent, the proportion who purchased it because they had seen it advertised on TV is 0.80. The marketing manager feels that decreases in the TV advertising budget over the years has caused a decrease in sales. She decides to conduct a survey to determine if there is sufficient evidence to conclude that the population proportion is less than 0.80. In a random sample of 400 consumers who bought this detergent, 296 indicated that they bought it because they saw it advertised on TV. What is the value of the test statistic for the test described above and what is the type of alternative hypothesis?

We are given that Historical data indicated that among the consumers who buy this detergent, the proportion who purchased it because they had seen it advertised on TV is 0.80. In a random sample of 400 consumers who bought this detergent, 296 indicated that they bought it because they saw it advertised on TV.

We have to test the hypothesis to determine if there is sufficient evidence to conclude that the population proportion is less than 0.80.

Let p = population proportion of people who purchased the detergent because they had seen it advertised on TV.

SO, Null Hypothesis, [tex]H_0[/tex] : p [tex]\geq[/tex] 0.80 {means that the population proportion of people who purchased the detergent because they had seen it advertised on TV is greater than or equal to 0.80}

Alternate Hypothesis, [tex]H_a[/tex] : p < 0.80 {means that the population proportion of people who purchased the detergent because they had seen it advertised on TV is less than 0.80}

The test statistics that will be used here is One-sample z-proportion test;

T.S. = [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ~ N(0,1)

where, [tex]\hat p[/tex] = proportion of people who bought detergent because they saw it advertised on TV in a sample of 400 customers = [tex]\frac{296}{400}[/tex] = 0.74

n = sample of customers = 400

SO, test statistics = [tex]\frac{0.74-0.80}{\sqrt{\frac{0.74(1-0.74)}{400} } }[/tex]

= -2.736

Therefore, the value of the test statistic for the test described above is -2.736.

And the type of alternative hypothesis is left-tailed because we have to determine whether the population proportion of people who purchased the detergent because they had seen it advertised on TV is less than 0.80.