Question 1171198: Suppose that historically the average age at which Canadians have been happier than at any other time in their lives is 37 years. A random sample of 50 Canadians taken in 2020 indicated that the mean happiest age was 40 years with a standard deviation equal to 16 years. At the 10 percent level of significance, does this sample indicate that there has been a change in the average age when Canadians are happiest?
a) State the null and alternative hypothesis.
b)State the decision rule
c)Compute the observed value of the test statistic.
d) What is your decision regarding the null hypothesis? Don’t forget your concluding statement.
e) Determine the p-value for this test.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! 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 ±zα/2 = ±z0.05.
* Using a standard normal distribution table or a calculator, we find that z0.05 ≈ 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.
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