Ardem collected data from a class survey. He then randomly selected samples of five responses to generate four samples.
| Survey Data |
|:----------|:---:|:---:|:---:|:---:|:---:|
| Sample 1 | 4 | 5 | 2 | 4 | 3 |
| Sample 2 | 2 | 2 | 6 | 5 | 7 |
| Sample 3 | 4 | 6 | 3 | 4 | 1 |
| Sample 4 | 5 | 2 | 4 | 3 | 6 |
Using his four samples, between what two numbers will Ardem's actual population mean lie?
A. 1 and 6
B. 2 and 5
C. 3.6 and 4.4
D. 4.0 and 4.4
Answer (1)
Calculate the mean of each sample: 3.6, 4.4, 3.6, 4.0.
Calculate the mean of the sample means: 3.9.
Determine the confidence interval using the t-critical value and SEM: approximately 3.37 to 4.43.
The actual population mean lies between \boxed{3.6 \text{ and } 4.4}.
Explanation
Calculate the Sample Means First, we need to calculate the mean of each sample. The means are calculated as follows:
Sample 1: 5 4 + 5 + 2 + 4 + 3 = 5 18 = 3.6 Sample 2: 5 2 + 2 + 6 + 5 + 7 = 5 22 = 4.4 Sample 3: 5 4 + 6 + 3 + 4 + 1 = 5 18 = 3.6 Sample 4: 5 5 + 2 + 4 + 3 + 6 = 5 20 = 4.0
Calculate the Mean of the Sample Means Next, we calculate the mean of the four sample means:
Mean of Means = 4 3.6 + 4.4 + 3.6 + 4.0 = 4 15.6 = 3.9
Calculate the Standard Deviation of the Sample Means Now, we calculate the standard deviation of the four sample means. The sample means are 3.6, 4.4, 3.6, and 4.0.
Standard Deviation of Means = 0.3316624790355401
Calculate the Standard Error of the Mean (SEM) We then calculate the standard error of the mean (SEM):
SEM = Number of Samples Standard Deviation of Means = 4 0.3316624790355401 = 2 0.3316624790355401 = 0.16583123951777004
Determine the t-critical value and calculate the Margin of Error Since the problem does not specify a confidence level, we will assume a 95% confidence interval. For a 95% confidence interval with 3 degrees of freedom (n-1 = 4-1 = 3), the t-critical value is approximately 3.182.
Now we calculate the margin of error:
Margin of Error = t-critical value * SEM = 3.182 * 0.16583123951777004 = 0.527675004145544
Construct the Confidence Interval Next, we construct the confidence interval:
Lower Bound = Sample Mean - Margin of Error = 3.9 - 0.527675004145544 = 3.3723249958544557 Upper Bound = Sample Mean + Margin of Error = 3.9 + 0.527675004145544 = 4.427675004145544
Therefore, the 95% confidence interval is approximately between 3.37 and 4.43.
Compare with the given options and conclude Comparing the calculated confidence interval (approximately 3.37 and 4.43) with the given options, the closest interval is 3.6 and 4.4.
Examples
In market research, analysts often collect data from multiple samples to estimate the average income of a population. By calculating the mean of each sample and then finding the confidence interval, they can determine a range within which the true population mean likely falls. For example, if four samples yield means of $30,000, $35,000, $32,000, and $33,000, the confidence interval might be $31,000 to $34,000, suggesting the true average income lies within this range with a certain level of confidence.
