The price for 2 tickets with online service charges, large popcorn and 2 medium soft drinks at a sample of 6 theater chains: $36.15, $31.00, $35.05, $40.25, $33.75, $43.00
a) Construct a 95% confidence interval estimate for the population mean the price for 2 tickets with online service charges, large popcorn and 2 medium soft drinks, assuming a normal distribution.
mean of sample = 36.53
n = 6
standard deviation = 19.26866 ( square root of ((36.15 - 36.53)^2 + (31 - 36.53)^2 + (35.05 - 36.53)^2 + (40.25 - 36.53)^2 + (33.75 - 36.53)^2 + (43 - 36.53)^2) / 6-1)
standard error of sample mean = 7.8664
critical value of t = 0.025
degrees of freedom = 5
95% C.I. = 36.53 +/- (critical value of t = 0.025)(7.8664)
95% C.I. = 36.53 +/- (2.571)(7.8664)
95% C.I. = 36.53 +/- 20.2245
My answer is wrong and the interval estimate is between 31.9267 and 41.1399. I think my error is solving for the standard deviation. How do you solve for it? Please provide a step-by-step solution. Thank you.
To calculate the standard deviation, follow these steps:
Step 1: Calculate the mean of the sample.
- Take the sum of all the observations and divide it by the number of observations.
- In this case, you have the values for 6 theater chains: $36.15, $31.00, $35.05, $40.25, $33.75, $43.00.
- Summing them gives: $36.15 + $31.00 + $35.05 + $40.25 + $33.75 + $43.00 = $219.20.
- Divide this sum by the number of observations (6) to get the mean: $219.20 / 6 = $36.53.
Step 2: Calculate the differences between each observation and the mean.
- Take each observation and subtract the mean calculated in step 1.
- For example, the difference between $36.15 and $36.53 is -$0.38.
Step 3: Square each difference calculated in step 2.
- Square each difference obtained in step 2, i.e., multiply each difference by itself.
- For example, the squared difference for -$0.38 is (-$0.38)^2 = $0.1444.
Step 4: Calculate the sum of the squared differences.
- Add up all the squared differences calculated in step 3.
- For all 6 observations, the sum is: $0.1444 + ... + (squared differences for other observations) = X.
Step 5: Divide the sum of squared differences by the degrees of freedom.
- Since you have 6 observations, the degrees of freedom will be (6 - 1) = 5.
- Divide the sum from step 4 by the degrees of freedom to get the variance.
- Variance = X / 5.
Step 6: Take the square root of the variance to find the standard deviation.
- Calculate the square root of the variance calculated in step 5 to get the standard deviation.
- Standard deviation = square root of (variance).
By following these steps, you should be able to accurately calculate the standard deviation to use in the confidence interval formula.