Lab Activity #_18__ Name(s)_Aya A. & Elizabeth S._

Probability and Sampling Distributions Date ___11/19/15_____

A recent Gallup Poll asked a simple random sample of 1600 American adults, “Have you, yourself smoked any

cigarettes in the past week?” Suppose that in fact 20% of all American adults would answer “yes” if asked this

question. The proportion of the sample who answers “yes” will vary in repeated sampling. To investigate

this, we simulated 1000 samples of size n = 1600 from a population in which 20% would answer “yes” they

smoked cigarettes in the past week. The results of this simulation are provided in the table below. Please fill in

the percentage column and write your answers as decimals rounded to three places.

pˆ

Frequency (of the 1000 samples) Percentage

0.165 to 0.170 1 0.1%

0.170 to 0.175 6 0.6%

0.175 to 0.180 22 2.2%

0.180 to 0.185 58 5.8%

0.185 to 0.190 89 8.9%

0.190 to 0.195 146 14.6%

0.195 to 0.200 178 17.8%

0.200 to 0.205 182 18.2%

0.205 to 0.210 160 16%

0.210 to 0.215 81 8.1%

0.215 to 0.220 49 4.9%

0.220 to 0.225 17 1.7%

0.225 to 0.230 7 0.7%

0.230 to 0.235 4 0.4%

Class for

pˆ

Please graph the distribution on the previous page with a histogram in the grid below. Place the classes of the

sample proportion on the x-axis (make each bar a width of one block) and the frequencies on the y-axis

(make each block a frequency of 10).

Once you have drawn the histogram, draw the density curve on the histogram that describes this distribution.

It turns out that the sampling distribution of for this scenario is a Normal distribution with mean 0.20 and

standard deviation 0.01. Use this information to answer the following questions:

Questions:

1. Find probability that at least 0.22 of the sample smokes:

a) Using the 68-95-99.7 Rule. Make sure to sketch the density curve and shade the area of interest.

z= (x-µ)/σ

.22-.20/.1=2 , using Table B percentile is 97.73

2.5%

pˆ

pˆ

b) Using the results of the simulation.

.017+ .007+.004= .028 (2.8%)

2. Find probability that fewer 0.19 of the sample smokes:

a) Using the 68-95-99.7 Rule. Make sure to sketch the density curve and shade the area of interest.

.19-.20/.01= -1 Making in it 15.87%

b) Using the results of the simulation.

.001+.006+.022+.058+.089=.176 (17.6%)

3. Find probability that between 0.18 and 0.22 of the sample smokes:

a) Using the 68-95-99.7 Rule. Make sure to sketch the density curve and shade the area of interest.

95%

b) Using the results of the simulation.

.058+.089+.146+.178+.182+.160+.081+.049= .943 (94.3%)

4. Are the results for parts a) and b) EXACTLY the same for questions #1, #2, #3 above?

No they’re not the same, but they’re close.

5. Why is there a difference between the results for parts a) and b) for each question above?

Because in part a, we estimated using the rule 68-95-99.7 or standardized scores formula, but but part b

adds up the percentages from the simulation accurately.

6. What would happen to the results for parts a) and b) if we simulated 50000 samples of size n = 1600 from a

population in which 20% would answer “yes” they smoked cigarettes in the past week.

The results would have a small change.

7. What statistical principle explains the answers to questions #5 and #6?

sampling distribution

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