According to a June 2009 report, 68% of people with “green” jobs in North America felt that they had job security, whereas 60% of people with green jobs in the United Kingdom felt that they had job security. Suppose that these results were based on samples of 305 people with green jobs from North America and 280 people with green jobs from the United Kingdom. Using the 2% significance level, can you conclude that proportion of all people with green jobs in North America who feel that they have job security is higher than the corresponding proportion for the United Kingdom?
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To determine whether the proportion of all people with green jobs in North America who feel they have job security is higher than the proportion for the United Kingdom, we can conduct a hypothesis test.
Here are the steps to perform the hypothesis test:
Step 1: State the hypotheses:
- Null hypothesis (H0): The proportion of people with green jobs in North America who feel they have job security is equal to or less than the proportion for the United Kingdom.
- Alternative hypothesis (Ha): The proportion of people with green jobs in North America who feel they have job security is higher than the proportion for the United Kingdom.
Step 2: Set the significance level:
The significance level, denoted as α, is the probability of making a Type I error (rejecting the null hypothesis when it is true). In this case, the significance level is given as 2%, which means we will reject the null hypothesis if the p-value is less than 0.02.
Step 3: Formulate the test statistic:
We will use the z-test statistic for comparing proportions.
The test statistic is calculated using the formula:
z = (p1 - p2) / sqrt((p̂ * (1 - p̂) / n1) + (p̂ * (1 - p̂) / n2))
where:
- p1: Sample proportion for North America
- p2: Sample proportion for the United Kingdom
- p̂: Pooled proportion (combined proportion of both samples)
- n1: Sample size for North America
- n2: Sample size for the United Kingdom
Step 4: Calculate the test statistic:
Using the given information:
- North America: p1 = 0.68, n1 = 305
- United Kingdom: p2 = 0.60, n2 = 280
To find the pooled proportion, calculate the weighted average of the sample proportions:
p̂ = (p1 * n1 + p2 * n2) / (n1 + n2)
Step 5: Determine the p-value:
The p-value is the probability of obtaining a test statistic as extreme as the observed, given that the null hypothesis is true. We will compare the p-value to the significance level to make our decision.
Step 6: Make a decision:
- If the p-value is less than the significance level (p-value < α), we reject the null hypothesis.
- If the p-value is greater than or equal to the significance level (p-value ≥ α), we fail to reject the null hypothesis.
So, by performing these steps, we can conclude whether the proportion of people with green jobs in North America who feel job security is higher than the proportion for the United Kingdom.