A random sample of n1=20n1=20 securities in Economy A produced mean returns of  x̄ 1=5.6% x̄ 1=5.6% with s1=2%s1=2% while another random sample of n2=17n2=17 securities in Economy B produced mean returns of  x̄ 2=4.6% x̄ 2=4.6% with s2=2.4%.s2=2.4%. At  α =0.02 α =0.02, can we infer that the returns differ significantly between the two economies?

Assume that the samples are independent and randomly selected from normal populations with **equal population variances** ( σ 12= σ 22)( σ 12= σ 22).

T-Distribution Table

**a.** Calculate the test statistic.

t=t= 

Round to three decimal places if necessary

**b.** Determine the critical value(s) for the hypothesis test.

 +

Round to three decimal places if necessary

**c.** Conclude whether to reject the null hypothesis or not based on the test statistic.

Reject

Fail to Reject

To test if the returns differ significantly between the two economies, we perform a two-sample t-test for the difference of means with equal variances.

### Hypotheses
- **Null Hypothesis (H₀)**: μ₁ - μ₂ = 0 (No difference in returns)
- **Alternative Hypothesis (H₁)**: μ₁ - μ₂ ≠ 0 (Returns are different)

### Given Data
- **Sample 1 (Economy A)**:
  - n₁ = 20
  - x̄₁ = 5.6%
  - s₁ = 2%
- **Sample 2 (Economy B)**:
  - n₂ = 17
  - x̄₂ = 4.6%
  - s₂ = 2.4%
- **Significance Level (α)**: 0.02

### Assumptions
- The samples are independent and randomly selected.
- The populations are normally distributed with equal variances (σ₁² = σ₂²).

#### (a) **Calculate the Test Statistic**

**Pooled Variance (sₚ²)**:
\[ s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \]

**Test Statistic (t)**:
\[ t = \frac{(x̄_1 - x̄_2) - (μ_1 - μ_2)}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \]

Calculations:

1. **Pooled Variance**:
   \[ s_p^2 = \frac{(20 - 1)(2)^2 + (17 - 1)(2.4)^2}{20 + 17 - 2} \]
   \[ s_p^2 = \frac{19(4) + 16(5.76)}{35} \]
   \[ s_p^2 = \frac{76 + 92.16}{35} \]
   \[ s_p^2 = \frac{168.16}{35} \]
   \[ s_p^2 \approx 4.80457 \]

2. **Pooled Standard Deviation**:
   \[ s_p = \sqrt{s_p^2} \]
   \[ s_p \approx \sqrt{4.80457} \]
   \[ s_p \approx 2.191 \]

3. **Test Statistic**:
   \[ t = \frac{(5.6 - 4.6) - 0}{2.191 \sqrt{\frac{1}{20} + \frac{1}{17}}} \]
   \[ t = \frac{1}{2.191 \sqrt{0.05 + 0.058824}} \]
   \[ t = \frac{1}{2.191 \sqrt{0.108824}} \]
   \[ t = \frac{1}{2.191 \times 0.32987} \]
   \[ t \approx \frac{1}{0.7233} \]
   \[ t \approx 1.382 \]

#### (b) **Determine the Critical Value(s) for the Hypothesis Test**

For a two-tailed test at α = 0.02, with degrees of freedom df = n₁ + n₂ - 2 = 35, we use the t-distribution table to find the critical value.

Critical value (t₀.01, 35) ≈ ±2.438 (obtained from t-distribution table for a two-tailed test with α/2 = 0.01)

#### (c) **Conclusion**

- **Test Statistic (t)**: 1.382
- **Critical Value**: ±2.438

Since the absolute value of the test statistic (1.382) is less than the critical value (2.438), we **fail to reject the null hypothesis**.

**Conclusion**: There is insufficient evidence at the 0.02 level of significance to conclude that the returns differ significantly between the two economies.

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Do you want more details or have any questions?

Here are 8 related questions:
1. What does it mean if we fail to reject the null hypothesis in this context?
2. How does the two-tailed nature of the test affect the interpretation of results?
3. What are the assumptions required for using the pooled variance t-test?
4. How would the conclusion change if the test were one-tailed?
5. What is the effect size, and how can it be calculated in this scenario?
6. What is the importance of checking for equal variances in the two samples?
7. Can this test be used if the data is not normally distributed? Why or why not?
8. How can the power of this test be increased?

**Tip:** Ensure that the sample sizes are sufficient to achieve a reliable conclusion, especially when working with small effect sizes and low significance levels.

Explore the two-sample t-test for mean returns in Economics A and B. Calculate the test statistic, determine critical values, and conclude hypothesis testing with detailed step-by-step solutions. Suitable for advanced college-level students and professionals.

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