Question

All parts please

1. A well-known brokerage firm executive claimed that at least 90% of investors are currently confident of meeting their investment goals. An XYZ Investor Optimism Survey, conducted over a two week period, found that in a sample of 800 people, 88% of them said they are confident of meeting their goals.

a) For this claim state the null and alternative hypothesis as an expression: (multiple choice)

A. H0:?=0.9

HA:??0.9

B. H0:p=0.9

HA:p<0.9

C. H0:?=0.9

HA:?>0.9

D. H0:?=0.9

HA:?<0.9

E. H0:p=0.9

HA:p>0.9

F. H0:p=0.9

HA:p?0.9

b) State the null hypothesis as a complete sentence:

State the alternative hypothesis as a complete sentence:

This test would require a _____ test. (multiple choice)

a. right-tailed

b. quad-tailed

c. left-tailed

d. two-tailed

c) How could we re-write the claim to make this a two-tail test? (sentence)

d) Test the claim that the proportion of people who are confident is smaller than 90% at the 0.05 significance level. Determine the following:

The test statistic:_____ (round to the 3rd decimal)

The p-value:_____ (round to the 4th decimal)

e) What would be our formal conclusion about the null hypothesis at a 0.05 significance level?

What practical conclusion can we make about the claim? (Answer as a complete sentence)

What would be our formal conclusion if we used a 0.01 significance level?

If you were a broker for this well know brokerage firm which significance level do you want the statistician to use and why? (Answer as a complete sentence)

If you were an investor shopping investment firms which significance level do you want the statistician to use and why? (Answer as a complete sentence)

How many people in the sample said they were confident of meeting their goals?

How many people were needed for this sample for 90% to have said they were confident of meeting their goals?

How many people would need to change their mind, how many people would need to go from saying they are not confident to saying they are confident, for 90% of the sample to say they are confident of meeting their goals?

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P > 0.90
Alternative hypothesis: P < 0.90

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation (S.D) and compute the z-score test statistic (z).

S.D = sqrt[ P * ( 1 - P ) / n ]

S.D = 0.01061
z = (p - P) / S.D

z = - 1.89

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a one-tailed test, the P-value is the probability that the z-score is less than -1.89

Thus, the P-value = 0.029

Interpret results. Since the P-value (0.029) is less than the significance level (0.05), we cannot accept the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that at least 90% of investors are currently confident of meeting their investment goals.