Central Limit Theorem (Cookie Recipes), The Central Limit Theorem | Statistics

Learning objectives

Central Limit Theorem (Cookie Recipes)

Student Learning Outcome

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 Data

Use a random number generator to randomly select four samples of size n = 5 from the given population. Record your samples in Ch07_lab2_tbl002. 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 following table:

	Sample 1	Sample 2	Sample 3	Sample 4





Means:	$\bar{x}$ = ____	$\bar{x}$ = ____	$\bar{x}$ = ____	$\bar{x}$ = ____

Calculate the following:
1. $\bar{x}$ = _______
2. s_$\bar{x}$ = _______

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 Ch07_lab2_tbl003. 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










Means:	$\bar{x}$ = ____	$\bar{x}$ = ____	$\bar{x}$ = ____	$\bar{x}$ = ____

Calculate the following:
1. $\bar{x}$ = ______
2. s_$\bar{x}$ = ______
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.

View original figure (external; image not reproduced in this text edition)

Original figure description: This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.
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.

Repeat the procedure for 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 $\frac{1}{2}$ day. Sketch the graph using a ruler and pencil. Scale the axes.
View original figure (external; image not reproduced in this text edition)
Original figure description: This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.
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.

Repeat the procedure for 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 $\frac{1}{2}$ day. Sketch the graph using a ruler and pencil. Scale the axes.

View original figure (external; image not reproduced in this text edition)

Original figure description: This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.
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.

Discussion Questions

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.
1. n = 5: $\bar{x}$ ~ _____(_____, _____)
2. n = 10: $\bar{x}$ ~ _____(_____, _____)
Are the sample means for n = 5 and n = 10 close to the theoretical mean, μ_x? Explain why or why not.
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.

Attribution. Statistics, OpenStax, Rice University, and the contributing authors credited in the book. Licensed under Creative Commons Attribution 4.0 International. Source revision 7dea80ae28 of openstax/osbooks-statistics (2026-04-23). CC BY 4.0. Original source ↗. Text edition of a pinned CC BY 4.0 source revision of the OpenStax book, converted from CNXML by the EvidenceHelix importer. Original prose, learning objectives, tables, exercises, glossary and captions are preserved; images are not copied (figures link to the original source and keep their credits). No new content was added. Later revisions of this book can carry different licence terms and are not used.

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

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

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