A. random sample of 50 four year olds attending daycare centers provided a yearly tuition average of 3987$ and the population standard deviation of $630. Find the 90% confidence interval of the true mean. and B. If a daycare center were starting up and wanted to keep tuition low, what would be a reasonable amount to charge?
90% = 3987 ± 1.645 (630)
How low would be reasonable would also depend on their start-up expenses, so I can't answer that.
To find the 90% confidence interval of the true mean for the tuition of four-year-olds attending daycare centers, we can use the formula:
Confidence Interval = Sample Mean ± (Z * (Standard Deviation / √(Sample Size))
where:
- Sample Mean is the average tuition of the random sample
- Z is the Z-score associated with the desired confidence level (90% in this case)
- Standard Deviation is the population standard deviation
- Sample Size is the number of observations in the random sample
Step A: Calculating the Confidence Interval
1. Substitute the given values into the formula:
Sample Mean = $3987
Standard Deviation = $630
Sample Size (n) = 50
Z-Score for 90% confidence level: You can look up the Z-score from a standard normal distribution table or use software. The Z-score for a 90% confidence level is approximately 1.645.
2. Calculate the standard error (SE) using the formula:
SE = Standard Deviation / √(Sample Size)
SE = $630 / √(50)
3. Calculate the margin of error (ME) using the formula:
ME = Z * SE
ME = 1.645 * ($630 / √(50))
4. Calculate the lower bound:
Lower Bound = Sample Mean - ME
5. Calculate the upper bound:
Upper Bound = Sample Mean + ME
The 90% confidence interval for the true mean tuition for four-year-olds attending daycare centers is [Lower Bound, Upper Bound].
Step B: Determining a reasonable tuition amount
If a daycare center wants to keep the tuition low, it can consider charging an amount that falls within the lower end of the 90% confidence interval. This would ensure that the tuition charged is not significantly higher than the true mean, with a 90% confidence level.