Question

A well-known brokerage firm executive claimed that at least 41 % of investors are currently confident of meeting their investment goals. An XYZ Investor Optimism Survey, conducted over a two week period, found that out of 88 randomly selected people, 35 of them said they are confident of meeting their goals. Suppose you have the following null and alternative hypotheses for a test you are running:

H0 : p = 0.41
H0 : p = 0.41
Ha : p < 0.41
Ha : p<0.41
Calculate the test statistic, rounded to 3 decimal places.

Answer 1

`z=\frac{0.398 -0.41}{\sqrt{(0.41(1-0.41))/(88)}}=-0.229`

Explanation:

1) Data given and notation n

n=88 represent the random sample taken

X=35 represent the adults that said they are confident of meeting their goals

`\hat p=(35)/(88)=0.398` estimated proportion of adults that said they are confident of meeting their goals

`p_o=0.41` is the value that we want to test

`\alpha` represent the significance level

z would represent the statistic (variable of interest)

`p_v` represent the p value (variable of interest)

2) Concepts and formulas to use

We need to conduct a hypothesis in order to test the claim that that at least 41 % of investors are currently confident of meeting their investment goals.:

Null hypothesis:
`p=0.41`

Alternative hypothesis:
`p < 0.41`

When we conduct a proportion test we need to use the z statisitc, and the is given by:

`z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}}` (1)

The One-Sample Proportion Test is used to assess whether a population proportion
`\hat p` is significantly different from a hypothesized value
`p_o`.

3) Calculate the statistic

Since we have all the info requires we can replace in formula (1) like this:

`z=\frac{0.398 -0.41}{\sqrt{(0.41(1-0.41))/(88)}}=-0.229`

4) Statistical decision

It's important to refresh the p value method or p value approach . "This methos is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.

The significance level is not provided, but we can assume
`\alpha=0.05`. The next step would be calculate the p value for this test.

Since is a unilateral test the p value would be:

`p_v =P(z<-0.229)=0.409`

Using the p value obtained and the significance level assumed
`\alpha=0.05` we have
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the proportion of of investors that are currently confident of meeting their investment goals is not significantly lower than 0.41 or 41%.