Question 1171198
Let's break down this hypothesis test step-by-step.

**a) State the Null and Alternative Hypotheses**

* **Null Hypothesis (H₀):** The average age at which Canadians are happiest is still 37 years.
    * H₀: μ = 37
* **Alternative Hypothesis (H₁):** The average age at which Canadians are happiest has changed (is not 37 years).
    * H₁: μ ≠ 37 (two-tailed test)

**b) State the Decision Rule**

* Significance level (α) = 10% = 0.10
* Sample size (n) = 50
* Since the population standard deviation is unknown and the sample size is large (n > 30), we will use a z-test.
* For a two-tailed test with α = 0.10, the critical z-values are ±z<sub>α/2</sub> = ±z<sub>0.05</sub>.
* Using a standard normal distribution table or a calculator, we find that z<sub>0.05</sub> ≈ 1.645.
* **Decision Rule:** Reject H₀ if the absolute value of the calculated z-statistic is greater than 1.645 (|z| > 1.645).

**c) Compute the Observed Value of the Test Statistic**

* Sample mean (x̄) = 40 years
* Population mean (μ) = 37 years
* Sample standard deviation (s) = 16 years
* Sample size (n) = 50

We use the z-statistic formula:

z = (x̄ - μ) / (s / √n)

z = (40 - 37) / (16 / √50)

z = 3 / (16 / 7.071)

z = 3 / 2.2627

z ≈ 1.326

**d) What is your Decision Regarding the Null Hypothesis?**

* The calculated z-statistic is 1.326.
* The critical z-values are ±1.645.
* Since |1.326| < 1.645, we fail to reject the null hypothesis.

**Concluding Statement:** There is not sufficient evidence at the 10% significance level to conclude that the average age when Canadians are happiest has changed from 37 years.

**e) Determine the p-value for this Test**

* The calculated z-statistic is 1.326.
* Since this is a two-tailed test, we need to find the probability of observing a z-statistic as extreme as 1.326 or -1.326.
* Using a standard normal distribution table or a calculator, we find the p-value:
    * P(Z > 1.326) ≈ 0.0924
    * P(Z < -1.326) ≈ 0.0924
    * p-value = 2 * 0.0924 ≈ 0.1848

* Since the p-value (0.1848) is greater than the significance level (0.10), we fail to reject the null hypothesis.