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 waking—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.

Question

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 waking—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 .05 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?)
Question- Sketch the distributions involved.

Answer 1

To determine the significance of the average cycle length differing from 24 hours, we need to conduct a statistical analysis using the provided data. Let's start by sketching the distributions involved.

The given data represents the cycle lengths of the eight individuals in the study after spending a month in a room without external cues. The cycle lengths are as follows: 25, 27, 25, 23, 24, 25, 26, and 25.

To sketch the distribution, we can create a histogram. We'll display the cycle lengths on the x-axis and the frequency (number of individuals) on the y-axis.

Here is a visualization of the data:

Cycle Length: Frequency:
23 *
24 *
25 ****
26 *
27 *

In this histogram, we can see the distribution of cycle lengths. The x-axis represents the possible values of the cycle length (ranging from 23 to 27), and the y-axis represents the frequency of each value.

The distribution appears to be centered around a cycle length of 25, as it has the highest frequency. The data also includes a few individuals with cycle lengths of 23, 24, 26, and 27.

Now, to determine whether the average cycle length significantly differs from 24 hours, we need to perform a statistical test, such as a one-sample t-test or a confidence interval. This test will provide a conclusion based on the given hypothesis and the significance level of 0.05.

Answer 2

To sketch the distributions involved, we can start by creating a histogram of the data provided, which represents the cycle lengths of the eight volunteers after spending a month in the room.

The cycle lengths are as follows: 25, 27, 25, 23, 24, 25, 26, and 25.

On the x-axis of the histogram, we will label the possible cycle lengths from 23 to 27, and on the y-axis, we'll represent the frequency or number of individuals who had each cycle length.

The histogram will have bars representing the frequencies for each cycle length. For example, if three individuals had a cycle length of 25, the bar representing 25 will be taller than the bars for other cycle lengths.

By observing the histogram, we can analyze the overall distribution of the cycle length data and make conclusions about whether the average cycle length significantly differs from 24 hours.