Calculate a 95% confidence interval for the mean income of all students given a mean income of $180.98 and a standard deviation of $143.04 from a survey of 8 university students.
Understand the Problem
The question is asking to calculate a 95% confidence interval for the mean income of all university students based on the survey data provided, which includes the mean income and standard deviation. To solve this, we will use the formula for a confidence interval.
Answer
$(L, U)$ where $L$ and $U$ are calculated as described in the steps above.
Answer for screen readers
The final answer should be expressed as an interval, in the form $(L, U)$, where $L$ is the lower bound and $U$ is the upper bound.
Steps to Solve
- Identify the necessary statistics
From the survey data, determine the following values needed to calculate the confidence interval: the sample mean ($\bar{x}$), the sample standard deviation ($s$), and the sample size ($n$).
- Determine the critical value
For a 95% confidence interval, we need to find the critical value $t^$ from the t-distribution table. You need to know the degrees of freedom, which is calculated as $df = n - 1$. Look up the corresponding $t^$ value for a 95% confidence level.
- Calculate the standard error (SE)
The standard error can be calculated using the formula: $$ SE = \frac{s}{\sqrt{n}} $$ where $s$ is the sample standard deviation and $n$ is the sample size.
- Calculate the margin of error (ME)
The margin of error can be found with the formula: $$ ME = t^* \times SE $$
- Construct the confidence interval
Finally, the confidence interval for the mean income can be constructed using the following formulas: $$ \text{Confidence Interval} = (\bar{x} - ME, \bar{x} + ME) $$ This gives the lower and upper bounds of the confidence interval.
The final answer should be expressed as an interval, in the form $(L, U)$, where $L$ is the lower bound and $U$ is the upper bound.
More Information
The confidence interval provides a range where we can expect the true mean income of all university students to fall 95% of the time. This is useful for making inferences based on sample data.
Tips
- Not using the appropriate $t^*$ value based on the degrees of freedom.
- Confusing the sample standard deviation with the population standard deviation.
- Not accounting for sample size when calculating the standard error.
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