The Central Limit Theorem

# Central Limit Theorem (Cookie Recipes)

OpenStaxCollege

[latexpage]

Class Time:

Names:

- The student will demonstrate and compare properties of the central limit theorem.

Given*X* = length of time (in days) that a cookie recipe lasted at the Olmstead Homestead. (Assume that each of the different recipes makes the same quantity of cookies.)

Recipe # | X |
Recipe # | X |
Recipe # | X |
Recipe # | X |
|||
---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 16 | 2 | 31 | 3 | 46 | 2 | |||

2 | 5 | 17 | 2 | 32 | 4 | 47 | 2 | |||

3 | 2 | 18 | 4 | 33 | 5 | 48 | 11 | |||

4 | 5 | 19 | 6 | 34 | 6 | 49 | 5 | |||

5 | 6 | 20 | 1 | 35 | 6 | 50 | 5 | |||

6 | 1 | 21 | 6 | 36 | 1 | 51 | 4 | |||

7 | 2 | 22 | 5 | 37 | 1 | 52 | 6 | |||

8 | 6 | 23 | 2 | 38 | 2 | 53 | 5 | |||

9 | 5 | 24 | 5 | 39 | 1 | 54 | 1 | |||

10 | 2 | 25 | 1 | 40 | 6 | 55 | 1 | |||

11 | 5 | 26 | 6 | 41 | 1 | 56 | 2 | |||

12 | 1 | 27 | 4 | 42 | 6 | 57 | 4 | |||

13 | 1 | 28 | 1 | 43 | 2 | 58 | 3 | |||

14 | 3 | 29 | 6 | 44 | 6 | 59 | 6 | |||

15 | 2 | 30 | 2 | 45 | 2 | 60 | 5 |

Calculate the following:

*μ*= _______

_{x}*σ*= _______

_{x}Collect the DataUse a random number generator to randomly select four samples of size *n* = 5 from the given population. Record your samples in [link]. Then, for each sample, calculate the mean to the nearest tenth. Record them in the spaces provided. Record the sample means for the rest of the class.

- Complete the table:

Sample 1 Sample 2 Sample 3 Sample 4 Sample means from other groups: Means: \(\overline{x}\) = ____ \(\overline{x}\) = ____ \(\overline{x}\) = ____ \(\overline{x}\) = ____ - Calculate the following:
- \(\overline{x}\) = _______
*s*_{\(\overline{x}\)}= _______

- Again, use a random number generator to randomly select four samples from the population. This time, make the samples of size
*n*= 10. Record the samples in [link]. As before, for each sample, calculate the mean to the nearest tenth. Record them in the spaces provided. Record the sample means for the rest of the class.

Sample 1 Sample 2 Sample 3 Sample 4 Sample means from other groups Means: \(\overline{x}\) = ____ \(\overline{x}\) = ____ \(\overline{x}\) = ____ \(\overline{x}\) = ____ - Calculate the following:
- \(\overline{x}\) = ______
*s*_{\(\overline{x}\)}= ______

- For the original population, construct a histogram. Make intervals with a bar width of one day. Sketch the graph using a ruler and pencil. Scale the axes.
- Draw a smooth curve through the tops of the bars of the histogram. Use one to two complete sentences to describe the general shape of the curve.

*n*= 5

- For the sample of
*n*= 5 days averaged together, construct a histogram of the averages (your means together with the means of the other groups). Make intervals with bar widths of \(\frac{1}{2}\) a day. Sketch the graph using a ruler and pencil. Scale the axes. - Draw a smooth curve through the tops of the bars of the histogram. Use one to two complete sentences to describe the general shape of the curve.

*n*= 10

- For the sample of
*n*= 10 days averaged together, construct a histogram of the averages (your means together with the means of the other groups). Make intervals with bar widths of \(\frac{1}{2}\) a day. Sketch the graph using a ruler and pencil. Scale the axes. - Draw a smooth curve through the tops of the bars of the histogram. Use one to two complete sentences to describe the general shape of the curve.

- Compare the three histograms you have made, the one for the population and the two for the sample means. In three to five sentences, describe the similarities and differences.
- State the theoretical (according to the clt) distributions for the sample means.
*n*= 5: \(\overline{x}\) ~ _____(_____,_____)*n*= 10: \(\overline{x}\) ~ _____(_____,_____)

- Are the sample means for
*n*= 5 and*n*= 10 “close” to the theoretical mean,*μ*? Explain why or why not._{x} - Which of the two distributions of sample means has the smaller standard deviation? Why?
- As
*n*changed, why did the shape of the distribution of the data change? Use one to two complete sentences to explain what happened.