Question

An accounting firm is considering offering investment advice in addition to its current focus on tax planning. Its analysis of the costs and benefits of adding this service indicate that it will be profitable if​ 40% or more of its current customer base use it. The firm plans to survey its customers. Let p denote the proportion of its customers who will use this service if​ offered, and let p denote the proportion who say in a survey that they will use this service. The firm does not want to invest in this expansion unless data show that it will be profitable.

Determine whether the following statement is true or false. If the statement is​ false, explain why it is false.

If the H0 ​holds, then p in the sample will be less than 0.4.

A. True
B. ​False, because if the H0 ​holds, then p in the sample will be greater than or equal to 0.4.
C. ​False, because of sampling variability.
D. ​False, because if the H0 is​ rejected, then p in the sample will be less than 0.4.

Answer 1

The assertion that if the H0 holds, then p in the sample will be less than 0.4 is false. The correct interpretation involves acknowledging sampling variability, meaning the sample proportion can vary around the true proportion due to chance. So the correct option is C.

Step-by-step explanation:

The statement “If the H0 holds, then p in the sample will be less than 0.4” is False. The null hypothesis (H0) in this context is likely set up to test whether the true proportion p of customers who will use the investment advice service is at least 40%. If we assume that H0: p ≥ 0.4, we would not expect p in the sample to be less than 0.4 specifically because of H0. p in the sample could be less than 0.4, equal to 0.4, or even more than 0.4 due to sampling variability and other factors affecting the sample. Hence, the correct answer is C – “False, because of sampling variability.”

Sampling variability acknowledges that there is a natural fluctuation in sample statistics from one sample to another. It is the randomness inherent in which samples of the customer base might say they will use the service as opposed to the true proportion of the entire customer base. Therefore, even if the true proportion (p) is 0.4 (as assumed under H0), p in the sample could still be less than 0.4, exactly 0.4, or more than 0.4 due to this variability.

Answer 2

(B) ​False, because if the
`H_(0)` ​holds, then p in the sample will be greater than or equal to 0.40.

Step-by-step explanation:

The analysis of the costs and benefits of adding this service shows that the it will be profitable if 40% or more of the firm's customer uses it.

Now to test this result the firm needs to survey its customers.

The hypothesis will be defined as:

`H_(0):` The proportion of customers using the service is 40% or more, i.e
`p\geq 0.40`

`H_(1):` The proportion of customers using the service is less than 40%, i.e
`p< 0.40`

Let X = number of customers who will use the service.

Then X will follow a Binomial distribution.

Using the Normal approximation we can approximate the binomial distribution by the normal distribution.

The test statistic used is:

`z=(\hat p-p)/(\sqrt(p(1-p))/(n) )`

If the null hypothesis (
`H_(0)`) is true, this implies that the test statistic is in the acceptance region. For this to happen the sample proportion (
`\hat p`) must be greater than the 0.40.

And as the test is left tailed the critical value will be negative.

So, with the test statistic is in the acceptance region and a negative critical value it will imply that the sample proportion is more than or equal to p = 0.40.

Thus, the correct option is (B).