Forty-six percent of people believe that there is life on other planets in the universe. A scientist does not agree with this finding. He surveyed 120 randomly selected individuals and found 48 believed that there is life on other planets. At α = 0.10, is there sufficient evidence to conclude that the percentage differs from 46?
To determine if there is sufficient evidence to conclude that the percentage differs from 46, we can conduct a hypothesis test.
Step 1: State the hypotheses:
- Null Hypothesis (H0): The percentage is equal to 46%
- Alternative Hypothesis (Ha): The percentage differs from 46%
Step 2: Set the significance level (α):
The significance level (α) is given as 0.10.
Step 3: Conduct the test and calculate the test statistic:
Using the given information, we can calculate the test statistic using the formula for a proportion:
Z = (p - P) / sqrt(P(1-P)/n)
where:
p = observed proportion (48/120)
P = hypothesized proportion (46% or 0.46)
n = sample size (120)
Plugging in the values, we get:
Z = (48/120 - 0.46) / sqrt(0.46*(1-0.46)/120)
Step 4: Calculate the critical value:
Since the alternative hypothesis is two-sided, we need to calculate the critical value using the significance level (α/2).
Using a Z-table or a statistical calculator, we find the critical value for α/2 = 0.05 to be approximately 1.645.
Step 5: Make a decision:
If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Conclusion:
Compare the test statistic to the critical value:
If |Z| > 1.645, reject the null hypothesis.
If |Z| ≤ 1.645, fail to reject the null hypothesis.
By calculating the test statistic and comparing it to the critical value, we can determine whether there is sufficient evidence to conclude that the percentage differs from 46.
To determine whether there is sufficient evidence to conclude that the percentage differs from 46%, we can perform a hypothesis test using the given information and the significance level α = 0.10.
Here are the steps to conduct the hypothesis test:
Step 1: Identify the null hypothesis (H0) and the alternative hypothesis (H1):
- Null hypothesis: The percentage of people who believe in life on other planets is equal to 46%.
- Alternative hypothesis: The percentage differs from 46%.
Step 2: Choose an appropriate test statistic and probability distribution:
Since we are analyzing the difference in proportions, we can use the normal distribution to approximate the sampling distribution of the difference.
Step 3: Set the significance level (α):
The significance level is given as α = 0.10.
Step 4: Calculate the test statistic:
In this case, we need to calculate the test statistic based on the sample data. The test statistic can be calculated using the formula:
\[ z = \frac{p - P}{\sqrt{\frac{P(1-P)}{n}}} \]
Where:
- p is the sample proportion (48/120 = 0.40),
- P is the hypothesized proportion (46/100 = 0.46), and
- n is the sample size (120).
Plugging in the values, we get:
\[ z = \frac{0.40 - 0.46}{\sqrt{\frac{0.46(1-0.46)}{120}}} \]
Calculating this will give us the value of the test statistic.
Step 5: Determine the critical value:
Since we have a two-tailed test (since the alternative hypothesis is that the percentage differs from 46%), we need to find the critical value(s) for α/2 = 0.05.
We can use a standard normal distribution table or a statistical software to find the critical value. For α = 0.05, the critical value is approximately ±1.96.
Step 6: Compare the test statistic with the critical value:
If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 7: Draw a conclusion:
Based on the comparison in Step 6, we can draw a conclusion about whether there is sufficient evidence to conclude that the percentage differs from 46%.
Please note that this explanation assumes knowledge of statistical hypothesis testing and the calculation of z-scores.