The average length of a roller coaster ride is 112 seconds with a known populati

Question

The average length of a roller coaster ride is 112 seconds with a known population standard deviation of 50.5 seconds. Suppose we take a sample of the last 100 roller coasters that were built and find that the average of those 100 is 105 seconds. We are interested in seeing if the actual average length of a roller coaster ride is significantly different than 112 seconds.

What is the standard error?

What is the margin of error at 90% confidence?

Using my sample of 100, what would be the 90% confidence interval for the population mean?

If I wanted to control my margin of error and set it to 5 at 90% confidence, what sample size would I need to take instead of the 100?

What are the null and alternative hypotheses?

What is the critical value at 90% confidence?

Calculate the test statistic (using the sample of 100).

Find the p-value.

What conclusion would be made here at the 90% confidence level?

Using the same data from question 1, it is known that only about 25% of roller coasters have inversions in them. Suppose I take the same sample of the last 100 roller coasters and find that only 15 out of 100 have inversions. I want to test to see if there is evidence that the number of rollercoasters with inversions is decreasing.

If I wanted to control my margin of error and set it to 5% with 99% confidence, what sample size would I need to take instead of the 100?

Using my original sample size of 100, what would be the 99% confidence interval for the population proportion?

What are the null and alternative hypotheses?

What is the critical value at 99% confidence?

Calculate the test statistic (using the sample of 100).

Find the p-value.

What conclusion would be made here at the 99% confidence level?

Explanation / Answer

mean = 112
std. dev. = 50.5
n = 100
xbar = 105

Standard Error = std.dev. / sqrt(n) = 50.5/sqrt(100) = 5.05

z-value = 1.645

ME = z*SE = 1.645*5.05 = 8.3073

Confidence Interval,
CI = (xbar - ME, xbar + ME)
CI = (103.69275, 120.30725)

Sample size, n = (z/ME * sigma)^2

n = (1.645/5 * 50.5)^2 = 276.04
Sample size = 276

H0: mu = 112
H1: mu not equals to 112

Critical values are (-1.645, 1.645)

Test statistics, t = (105 - 112)/5.05 = -1.3861

p-value = 0.0829

Failed to reject the null hypothesis.

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