Question

A group of consumers were randomly selected and asked whether they were planning to buy a new TV screen within the next year. 56 said "yes," and 174 said "no." Suppose the true population proportion of consumers that plan to buy a new TV screen within the next year is 0.22. Use a technique learned in class to approximate the probability of obtaining a sample proportion equal to or larger than the one obtained here. The approximate probability is _______.

Answer 1

Answer:

The approximate probability is 0.1921.

Step-by-step explanation:

The p-value is defined as the probability, [under the null-hypothesis (H₀)], of attaining a result equivalent to or greater than what was the truly observed value of the test statistic.

The hypothesis to test whether the proportion of consumers that plan to buy a new TV screen within the next year is 0.22, is defined as:

H₀: The proportion of consumers that plan to buy a new TV screen within the next year is 0.22, i.e. p = 0.22.

Hₐ: The proportion of consumers that plan to buy a new TV screen within the next year is 0.22, i.e. p > 0.22.

The information provided is:

n = 230

X = 56

Compute the value of sample proportion as follows:

$\hat p=\frac{X}{n}=\frac{56}{230}=0.2435$

Compute the test statistic as follows:

$z=\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}}}=\frac{0.2435-0.22}{\sqrt{\frac{0.22(1-0.22)}{230}}}=0.87$

The test statistic value is, 0.87.

Compute the p-value as follows:

$p-value=P(Z>0.87)=1-P(Z<0.87)=1-0.8079=0.1921$

*Use a z-table for the probability.

The p-value of the test is 0.1921.

Thus, the approximate probability of obtaining a sample proportion equal to or larger than the one obtained here is 0.1921.