SOLUTION: A well-known brokerage firm executive claimed that 40% of investors are currently confident of meeting their investment goals. An XYZ Investor Optimism Survey, conducted over a tw

Question 1183081: A well-known brokerage firm executive claimed that 40% 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 700 people, 31 of them said they are confident of meeting their goals.
Test the claim that the proportion of people who are confident is smaller than 40% at the 0.25 significance level.
a) The null and void would be:
b) The test statistic is ____ (to 3 decimals)
c) the p-value is: _____ ( to 4 decimals )

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part (a)

I think you meant to say "null and alternative hypothesis". I'm not familiar with the term "void" when it comes to statistical hypotheses.

The null would be p = 0.40 which is based on previous studies.

The alternative hypothesis is p < 0.40

We can use this shorthand notation
H0: p = 0.40
H1: p < 0.40
to represent the null and alternative hypothesis

Based on the inequality sign of the alternative hypothesis, we have a left tailed test.

=======================================================================

Part (b)

x = number of successes
x = number of people confident in reaching their goals
x = 31

n = sample size
n = number of people asked in the survey
n = 700

phat = sample proportion
This value is pronounced as "p-hat" though I'll leave out the minus sign to avoid confusion later on.

phat = x/n
phat = 31/700
phat = 0.0442857 approximately

SE = standard error
SE = sqrt(p*(1-p)/n)
SE = sqrt(0.40*(1-0.40)/700)
SE = 0.0185164 also approximate

We have enough to compute the z test statistic
z = (phat - p)/SE
z = (0.0442857 - 0.40)/0.0185164
z = -19.210769912078

The test statistic is approximately -19.211

As you'll see in the next part, this isn't good.
=======================================================================

Part (c)

Because the test statistic is very far away from the interval -3 < z < 3, this means that the p-value is effectively zero.

In other words, P(Z < -19.211) = 0
It's technically not exactly equal to 0, but it's so very very small that it might as well be treated as zero.

I'm not sure if your teacher intended this. There might be a typo and that would mean the result of part (b) should be something else. Ideally we want a z value somewhere between z = -3 and z = 3.

If I had to guess, it's possible that the "700" should be "70". Or perhaps the "31" should be much larger. I would ask your teacher for clarification.

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