A group of 49 randomly selected construction workers has a mean age of 22.4 years with a standard deviation of 3.8. According to a recent survey, the mean age should be mu=21.9. Test this hypothesis by constructing a 98% confidence interval for the population mean.
To construct a 98% confidence interval for the population mean, we can follow these steps:
Step 1: Identify the necessary information:
- Sample mean (x̄) = 22.4
- Sample size (n) = 49
- Standard deviation (s) = 3.8
- Confidence level (C) = 98%
Step 2: Calculate the standard error (SE):
SE = s / √n
SE = 3.8 / √49
SE = 3.8 / 7
SE = 0.5428571428571428
Step 3: Determine the critical value:
For a 98% confidence interval, we need to find the critical value that leaves 1% in each tail. Since the sample size is greater than 30, we can use the z-distribution. The critical value can be found using the z-table or a calculator.
The z-value for a 1% tail on each side is 2.57 (approximately).
Step 4: Calculate the margin of error (ME):
ME = critical value (z) * standard error (SE)
ME = 2.57 * 0.5428571428571428
ME = 1.39324
Step 5: Calculate the lower and upper limits of the confidence interval:
Lower limit = Sample Mean - Margin of Error
Upper limit = Sample Mean + Margin of Error
Lower limit = 22.4 - 1.39324
Lower limit = 21.00676
Upper limit = 22.4 + 1.39324
Upper limit = 23.80676
Step 6: Interpret the confidence interval:
The 98% confidence interval for the population mean age is (21.00676, 23.80676). This means that we are 98% confident that the true population mean age lies between these two values. Since the hypothesized mean age (mu=21.9) falls within this interval, we cannot reject the null hypothesis that the mean age is equal to 21.9.
To construct a confidence interval for the population mean and test the given hypothesis, we can follow these steps:
Step 1: Identify the information given:
- Sample size (n) = 49
- Sample mean (x̄) = 22.4
- Population mean (μ) from the survey = 21.9
- Standard deviation (σ) = 3.8
- Confidence level = 98%
Step 2: Determine the critical value:
Since the confidence level is 98%, we need to find the critical value with alpha (α) = 1 - (confidence level/100) = 1 - 0.98 = 0.02.
Since the sample size is large (n > 30), we can assume the sampling distribution is approximately normal and use the z-distribution.
Look up the critical value (z) associated with a two-tailed test for significance level alpha/2 = 0.02/2 = 0.01. From a standard normal distribution table, the critical z-value is approximately 2.33.
Step 3: Calculate the margin of error (E):
The margin of error is given by E = z * (σ / √n), where z is the critical value, σ is the population standard deviation, and n is the sample size.
E = 2.33 * (3.8 / √49) ≈ 1.42
Step 4: Calculate the confidence interval:
The confidence interval can be calculated as (x̄ - E, x̄ + E), where x̄ is the sample mean and E is the margin of error.
Confidence interval = (22.4 - 1.42, 22.4 + 1.42)
Confidence interval = (20.98, 23.82)
Step 5: Evaluate the hypothesis:
To test the given hypothesis (μ = 21.9) using the confidence interval, we can check if the hypothesized value (21.9) falls within the confidence interval.
Since the hypothesized value of 21.9 falls within the confidence interval of (20.98, 23.82), we fail to reject the null hypothesis.
Conclusion:
Based on the construction worker sample, with a 98% confidence level, the data does not provide sufficient evidence to conclude that the mean age is significantly different from the population mean of 21.9 years.
21. 9 -2.33* 3.8 /sqrt((49)), 21. 9 + 2.33* 3.8 /sqrt((49))
(20.64, 23.16)
98% confidence interval contain