According to the New York Stock Exchange, the mean portfolio value for U.S. senior citizens who are shareholders is $183,000. Assume portfolio values are normally distributed. Suppose a simple random sample of 51 senior citizen shareholders in a certain region of the United States is found to have a mean portfolio value of $198,000, with a standard deviation of $65,000.
a. From these sample results, and using the 0.05 level of significance comment on whether the mean portfolio value for all senior citizen shareholders in this region might not be the same as the mean value reported for their counterparts across the nation, by using the critical value method. Establish the null and alternative hypotheses.
b. What is your conclusion about the null hypothesis?

The mean portfolio value for all senior citizen shareholders in this region might not be the same as the mean value reported for their counterparts across the nation

Step-by-step explanation:

Step(i):-

Given mean of the population (μ) = $183,000

Given mean of the sample (x⁻) = $198,000

Given standard deviation of the sample (S) = $65,000.

Mean of the sample size 'n' = 51

level of significance α = 0.05

Step(ii):-

Null hypothesis : H₀ : There is no significance difference between the means

Alternative Hypothesis :H₁: There is significance difference between the means

Test statistic

[tex]t = \frac{x^{-}-mean }{\frac{S.D}{\sqrt{n} } }[/tex]

[tex]t = \frac{198,000- 183,000 }{\frac{ 65,000}{\sqrt{51} } }[/tex]

t = 1.64

Step(iii)

Degrees of freedom ν = n-1 = 51-1 =50

t₀.₀₅ = 2.0086

The calculated value t = 1.64 < 2.0086 at 0.05 level of significance

Null hypothesis is accepted

Final answer:-

There is no significance difference between the means

The mean portfolio value for all senior citizen shareholders in this region might not be the same as the mean value reported for their counterparts across the nation