Answer:
Joe Mama can conclude that there is significant evidence to conclude that more than 70% of the total population of customers will be interested in buying into the new investment opportunity.
Explanation:
We can check for the right conclusion from the results of this sample with an hypohesis test or finding the confidence interval of the population proportion that would be willing to buy into the new investment opportunity.
Using the hypothesis test method
For hypothesis testing, the first thing to define is the null and alternative hypothesis.
The null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.
While, the alternative hypothesis usually confirms the the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test.
For this question, the null hypothesis is that there is no significant evidence to conclude that more than 70% of the total population of customers will be interested in buying into the new investment opportunity.
The alternative hypothesis will now be that there is significant evidence to conclude that more than 70% of the total population of customers will be interested in buying into the new investment opportunity.
Mathematically,
The null hypothesis is represented as
H₀: p ≤ 0.70
The alternative hypothesis is given as
Hₐ: p > 0.70
To do this test, we will use the t-distribution because no information on the population standard deviation is known
So, we compute the t-test statistic
t = (x - μ)/σₓ
x = p = sample proportion that are interested in buying into.the new investment opportunity = (117/150) = 0.78
μ = p₀ = the benchmark we're comparing against = 0.70
σₓ = standard error = √[p(1-p)/n]
where n = Sample size = 150
σₓ = √[0.78×0.22/150] = 0.0338
t = (0.78 - 0.70) ÷ 0.0338
t = 2.37
checking the tables for the p-value of this t-statistic
Degree of freedom = df = n - 1 = 150 - 1 = 149
Significance level = 0.05 (most tests are performed at this significance level)
The hypothesis test uses a one-tailed condition because we're testing only in one direction.
p-value (for t = 2.37, at 0.05 significance level, df = 149, with a one tailed condition) = 0.009534
The interpretation of p-values is that
When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.
So, for this question, significance level = 0.05
p-value = 0.009534
0.009534 < 0.05
Hence,
p-value < significance level
This means that we reject the null hypothesis, accept the alternative hypothesis & say that there is significant evidence to conclude that more than 70% of the total population of customers will be interested in buying into the new investment opportunity.
Hope this Helps!!!