Evolutionary theories often emphasize that humans have adapted to their physical environment. One such theory hypothesizes that people should spontaneously follow a 24- hour cycle of sleeping and walking- even if they are not exposed to the usual pattern of sunlight. To test this notion, eight paid volunteers were placed (individually) in a room in which there was no light from the outside and no clocks or other indications of time. They could turn the lights on and off as they wished. After a month in the room, each individual tended to develop a steady cycle. Their cycles at the end of the study were as follows: 25, 27, 25, 23, 24, 25, 26, and 25
Using the 5% level of significance, what should we conclude about the theory that 24 hours is the natural cycle? (That is, does the average cycle length under these conditions differ significantly from 24 hours?)
(a) Use the steps of hypothesis testing.
(b) Sketch the distributions involved.
(c) Explain your answer to someone who has never taken a course in statistics.
You can probably use a one-sample t-test for this data. You'll need to calculate the mean and standard deviation. Here's a t-test formula:
t = (sample mean - population mean)/(standard deviation divided by the square root of the sample size)
Hypotheses:
Ho: µ = 24 -->null hypothesis
Ha: µ does not equal 24 -->alternate hypothesis
Once you do the calculations, use a t-table to determine the cutoff to reject the null at .05 level of significance for a two-tailed test using 7 degrees of freedom (df = n - 1 = 8 - 1 = 7).
Does the test statistic exceed the cutoff (critical) value from the table? If it does not, you cannot reject the null and conclude a difference. If it does, then reject the null and conclude a difference.
I hope this will help.
(a) The steps of hypothesis testing are as follows:
Step 1: Formulate the null hypothesis (H0) and alternative hypothesis (Ha):
H0: The average cycle length is 24 hours.
Ha: The average cycle length is not 24 hours.
Step 2: Set the significance level (α):
Given the significance level of 5% (α = 0.05).
Step 3: Collect data and calculate the test statistic:
Calculate the sample mean of the observed cycle lengths and determine the standard deviation. Then use this information to calculate the t-test statistic.
Step 4: Determine the critical value(s):
For a two-tailed test at a 5% level of significance, we divide α by 2 (α/2 = 0.025) and use the t-distribution table to find the critical t-value(s) for the appropriate degrees of freedom.
Step 5: Make a decision:
If the absolute value of the test statistic falls outside the range of the critical t-values, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
(b) The sketch of the distribution involved in this scenario would be a t-distribution with degrees of freedom n-1, where n is the number of samples.
(c) We want to determine if the average cycle length under these conditions differs significantly from 24 hours. To test this, we conducted a study where eight paid volunteers were placed in a room without any indications of time for a month. Their observed cycle lengths were recorded and analyzed.
Using hypothesis testing with a 5% level of significance, we formulated a null hypothesis that the average cycle length is 24 hours. The alternative hypothesis states that the average cycle length is different from 24 hours.
We calculated the t-test statistic using the observed data and determined the critical t-value(s) from the t-distribution table based on the degrees of freedom. If the absolute value of the test statistic falls outside the range of the critical t-values, we reject the null hypothesis and conclude that there is evidence that the average cycle length differs significantly from 24 hours. Otherwise, if the test statistic falls within the range of the critical t-values, we fail to reject the null hypothesis.
(a) To conduct hypothesis testing, we will follow these steps:
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha)
- Null Hypothesis (H0): The average cycle length under these conditions is 24 hours.
- Alternative Hypothesis (Ha): The average cycle length under these conditions differs significantly from 24 hours.
Step 2: Determine the appropriate statistical test
- Since we are comparing the mean cycle length to a specific value, we will perform a one-sample t-test.
Step 3: Set the significance level (α)
- The significance level is given as 5% or 0.05.
Step 4: Collect the data and calculate the test statistic
- Given the cycle lengths of the eight individuals: 25, 27, 25, 23, 24, 25, 26, 25.
- Calculate the sample mean (x̄) and sample standard deviation (s).
Step 5: Determine the critical value or p-value
- Using the calculated test statistic, determine the p-value or compare it with the critical value from the t-distribution table.
Step 6: Make a decision
- If the p-value is less than the significance level, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
(b) To sketch the distributions involved, we need to consider the null hypothesis that the average cycle length is 24 hours. We can assume a normal distribution around the mean value of 24 and sketch the probability density function curve.
(c) If you haven't taken a course in statistics, here's an explanation in simple terms:
- The researchers conducted an experiment with eight volunteers to test whether people spontaneously follow a 24-hour cycle of sleeping and waking up, even without exposure to the usual pattern of sunlight.
- The null hypothesis assumes that the average cycle length is 24 hours, while the alternative hypothesis suggests that it differs significantly from 24 hours.
- To test this, they collected data on the cycle lengths of the participants after a month in a room with no outside light or indications of time.
- Using statistical analysis, they calculated the sample mean and standard deviation of the cycle lengths and performed a one-sample t-test to compare it to 24 hours.
- Based on the analysis, they determined the p-value, which represents the likelihood of observing the results if the null hypothesis were true.
- The significance level, set at 5%, helps in deciding whether the difference between the observed cycle lengths and 24 hours is statistically significant.
- If the p-value is less than 0.05, they can conclude that the average cycle length significantly differs from 24 hours, rejecting the null hypothesis. Otherwise, if the p-value is higher than 0.05, they fail to reject the null hypothesis, suggesting that there is not enough evidence to conclude a significant difference from 24 hours.