Question

1. The average annual total return for U.S. Diversified Equity mutual funds from 1999 to 2003 was 4.1%. A researcher would like to conduct a hypothesis test to see whether the returns for mid-cap growth funds over the same period are significantly different from the average for U.S. Diversified Equity funds. A sample of 40 midcap growth funds provides a mean return of 3.4%. Assume that the population standard deviation for mid-cap growth funds is known from previous studies to be 2%. 1A At the 0.05 level of significance, using the Critical Value Using Z Method, determine whether the mean annual return for mid-cap growth funds differs from the mean for U.S. Diversified Equity funds. Make sure that you state the null and alternative hypotheses. Do the following steps to answer A & use the Critical Value Using Z Method:
   
   a. State the Null and Alternative hypotheses.

Ho:
Ha:
Do we have a two-tail, left-tail or right tail test? How does that affect our work?

b. What is the hypothesized mean µo?
  

c. Do we know σ : Yes or No ? Yes

What does that mean?

What is the value of σ? 2

d.What is α?
What is another name for α?

e. What is the definition of a Type I Error?

f. What is the value of α/2?

g. Find Zα/2
Same process as with Confidence Intervals

h. Find the test Statistic Z*
Find the Lower Critical Value of Z
LCL= Lowest Critical Value Find the Upper Critical Value of Z
UCV=Upper Critical Value

i. Draw the Bell curve indicating the Rejection Regions and the Non-Rejection Region, and then place the Critical Values and the Z* Test Statistic on the Z line.

j. State your decision:

Answer 1

Answer: The average annual total return for U.S. Diversified Equity mutual funds from 1999 to 2003 was 4.1%. A researcher would like to conduct a hypothesis test to see whether the returns for mid-cap growth funds over the same period are significantly different from the average for U.S. Diversified Equity funds. A sample of 40 midcap growth funds provides a mean return of 3.4%. Assume that the population standard deviation for mid-cap growth funds is known from previous studies to be 2%.

Solution:

a. State the Null and Alternative hypotheses.

Ho: μ = 4.1

Ha: μ ≠ 4.1

b. The hypothesized mean µo = 4.1

c. Do we know σ : Yes

What does that mean?

The population standard deviation 2 indicates the return for mid-cap growth funds around the mean x̄ = 3.4

What is the value of σ^2

σ^2 = 22 = 4

d. What is α?

α is the significance level.

α = 0.05

What is another name for α?

The another name for α is Type I Error.

e. What is the definition of a Type I Error?

A Type I error occurs if you rejects the null hypothesis Ho when it is true and should not be rejected. The probability of committing a Type I error is called the significance level and is often denoted by α.

f. What is the value of α/2?

α/2 = 0.05/2 = 0.025

g. Find Zα/2

Zα/2 = 1.96

Same process as with Confidence Intervals

h. Find the test Statistic, Z:

Z = x̄ - μ / σ/√n

Z = 3.4 - 4.1 / 2/√40

Test statistic, Z = - 2.2135

Find the Lower Critical Value and Upper Critical Value of Z:

LCV = lowest critical value = - 1.96

UCV = upper critical value = + 1.96

i. Draw the Bell curve indicating the Rejection Regions and the Non-Rejection Region, and then place the Critical Values and the Z* Test Statistic on the Z line.

Rejection region:

Reject Ho if Z > +1.96

or if Z < -1.96

otherwise do not reject Ho.

j. State your decision:

Since, Z (-2.2135) < - 1.96

We reject the null hypothesis, Ho.

Therefore, there is enough evidence to claim that the returns for mid-cap growth funds over the same period are significantly different from the average for U.S. Diversified Equity funds. , at the 0.05 significance level.

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