# 7.5 Central limit theorem (cookie recipes)

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Class Time:

Names:

## Student learning outcomes

• 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:

1. μ x = _______
2. 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 [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.

1. Complete the table:
Sample 1 Sample 2 Sample 3 Sample 4 Sample means from other groups:
Means: $\overline{x}$ = ____ $\overline{x}$ = ____ $\overline{x}$ = ____ $\overline{x}$ = ____
2. Calculate the following:
1. $\overline{x}$ = _______
2. s $\overline{x}$ = _______
3. 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: $\overline{x}$ = ____ $\overline{x}$ = ____ $\overline{x}$ = ____ $\overline{x}$ = ____
4. Calculate the following:
1. $\overline{x}$ = ______
2. s $\overline{x}$ = ______
5. 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.
6. 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

1. 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.
2. 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

1. 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.
2. 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

1. 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.
2. State the theoretical (according to the clt) distributions for the sample means.
1. n = 5: $\overline{x}$ ~ _____(_____,_____)
2. n = 10: $\overline{x}$ ~ _____(_____,_____)
3. Are the sample means for n = 5 and n = 10 “close” to the theoretical mean, μ x ? Explain why or why not.
4. Which of the two distributions of sample means has the smaller standard deviation? Why?
5. As n changed, why did the shape of the distribution of the data change? Use one to two complete sentences to explain what happened.

binomial distribution tends to normal distribution
if sample size n very large and probability tends to 0.5 if these both conditions are satisfied by binomial distribution it would tends to normal distribution
sravani
n tends to infinite i.e large Probability tends to 0 i.e indefinitely small. np = lamda
Anji
the above is poison to Bin
Anji
no of trails n tends to indefinitely large..i.e infine neither p nor q is very small Then bin tends to normal
Anji
if the death of of the snow is my yard is normally distributed with the m is equals to 2.5 and what is the probability that a randomly chosen location with have a no that between 2.25 and 2.76
hey
Shubham
🤔
Iqra
hello
Sakshi
hii
Rushikesh
helow
why Statistics so hard
Mohd
ho geya solve
Sakshi
it's not hard
Sakshi
it is hard 😭
Mohd
solution?
Abdul
hii
it's just need to be concentrate
Akinyemi
exactly..... concentration is very important
Iqra
rewrite the question
what is the true statement about random variable?
A consumer advocate agency wants to estimate the mean repair cost of a washing machine. the agency randomly selects 40 repair cost and find the mean to be $100.00.The standards deviation is$17.50. Construct a 90% confidence interval for the mean.
pls I need understand this statistics very will is giving me problem
Sixty-four third year high school students were given a standardized reading comprehension test. The mean and standard deviation obtained were 52.27 and 8.24, respectively. Is the mean significantly different from the population mean of 50? Use the 5% level of significance.
No
Ariel
how do I find the modal class
look for the highest occuring number in the class
Kusi
the probability of an event occuring is defined as?
The probability of an even occurring is expected event÷ event being cancelled or event occurring / event not occurring
Gokuna
what is simple bar chat
Simple Bar Chart is a Diagram which shows the data values in form of horizontal bars. It shows categories along y-axis and values along x-axis. The x-axis displays above the bars and y-axis displays on left of the bars with the bars extending to the right side according to their values.
statistics is percentage only
the first word is chance for that we use percentages
it is not at all that statistics is a percentage only
Shambhavi
I need more examples
how to calculate sample needed
mole of sample/mole ratio or Va Vb
Gokuna
how to I solve for arithmetic mean
Yeah. for you to say.
James
yes
niharu
how do I solve for arithmetic mean
niharu
add all the data and divide by the number of data sets. For example, if test scores were 70, 60, 70, 80 the total is 280 and the total data sets referred to as N is 4. Therfore the mean or arthritmatic average is 70. I hope this helps.
Jim
*Tan A - Tan B = sin(A-B)/CosA CosB ... *2sinQ/Cos 3Q = tan 3Q - tan Q