The average cost per night of a hotel room in New York City is $273 (SmartMoney, March 2009). Assume this estimate is based on a sample of 45 hotels and that the sample standard deviation is $65.
a. With 95% confidence, what is the margin of error (to 2 decimals)?
b. What is the 95% confidence interval estimate of the population mean?
NOTE: This is what I got but it is wrong.
a. 18.99
b. [254.008, 291.992]
z table
a) 18.99167069
b) (254, 292)
To calculate the margin of error and confidence interval, we will use the formula:
Margin of Error (E) = Z * (SD / sqrt(n))
Where:
Z = Z-score (corresponding to the desired confidence level)
SD = Sample standard deviation
n = Sample size
a. To find the margin of error:
Given: Z-score for a 95% confidence level is 1.96 (from the standard normal table)
SD = $65
n = 45
Plug the values into the formula:
E = 1.96 * (65 / sqrt(45))
E ≈ 1.96 * (65 / 6.708)
E ≈ 1.96 * 9.69
E ≈ 18.98 (rounded to 2 decimal places)
Therefore, the margin of error is approximately $18.98.
b. To find the confidence interval estimate of the population mean:
Using the margin of error, we can calculate the upper and lower bounds of the confidence interval.
Lower Bound = Sample mean - Margin of Error
Upper Bound = Sample mean + Margin of Error
Given: The average cost per night of a hotel room in New York City is $273 (sample mean)
Margin of Error (E) = $18.98
Lower Bound = $273 - $18.98 = $254.02
Upper Bound = $273 + $18.98 = $292.98
The 95% confidence interval estimate of the population mean is approximately [$254.02, $292.98].
(Note: The interval you provided [254.008, 291.992] is very close but seems to have been rounded incorrectly.)
To calculate the margin of error and the confidence interval estimate, you can use the formula:
Margin of Error (ME) = Critical Value * (Standard Deviation / √(Sample Size))
Confidence Interval = Sample Mean ± Margin of Error
Let's calculate each step separately:
a. Calculating the Margin of Error:
First, we need to find the critical value associated with a 95% confidence level. Since the sample size is 45 (which is relatively large), we can use the z-score table for a normal distribution. The z-score for a 95% confidence level is 1.96.
ME = 1.96 * (65 / √(45))
ME ≈ 18.99 (rounding to two decimal places)
Therefore, the margin of error is approximately 18.99.
b. Calculating the Confidence Interval:
The confidence interval can be calculated using the formula:
Confidence Interval = Sample Mean ± Margin of Error
The sample mean is given as $273.
Confidence Interval = $273 ± 18.99
Confidence Interval = [$254.01, $291.99]
Therefore, the 95% confidence interval estimate of the population mean is approximately [$254.01, $291.99].