Question

A survey was done: Do you favor Soap XX? 1 = Yes 0 = No and the results were as follows

1 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 0 1

a) How many people said YES and how many people said NO?

b) What is the sample size?

c) What is the proportion of people in the sample that said YES?

d) Form the 93% confidence interval for the proportion of people that FAVOR Soap XX (Said YES).

e) Use Hypothesis Testing (P-Value Method) for the claim that the proportion of people that FAVOR Soap XX is greater than 50% at 90% confidence level.

f) What is your conclusion for the Hypothesis Test, state it very clearly?

Answer 1

From the Given problem ,

a) No. of people who said YES ( 1 = YES ) = 22

No. of people who said No ( 0 = NO ) = 18

b) Sample size = 40

c) Proportion of people who said YES = 22 / 40 = 0.55

d) 93 % Confidence interval for proportion is

From CDF of normal distribution     

  

C.I = (0.4076, 0.6924)

f )   Final answers are given below.back-up Theory and Details of Calculations follow at the end.

i) The claim that the proportion of people that FAVOR soap XX is greater than 50% at 90% confidence level is NOT valid.

ii ) Conclusion : Less than 50% of the population FAVOR soap XX.

Back-up Theory and Details of Calculations :

Let X = of people out of 30 that FAVOR Soap XX

Then ,

  

Where , n = 40 = sample size

p = 0.5 (i.e., 50%) = proportion that a person FAVOR Soap XX , which is also equal to the population proportion.

  Claim : Number of people that FAVOR Soap XX is greater than 50%.

Hypotheses :

Null   : p = po = 0.5 Vs Alternative   : p > 0.5 (claim)

Test Statistic :

Where

n = sample size.

Calculations :

Given	P > Po
P0	0.5
n	40
X	22
	0.55
	0.6325
	0.1
	1.28156
p - value	0.2635

Distribution , Significance level , , Critical value and p - value :

Under H0, disribution of Z can be approximated by Standard Normal Disribution, provided

and are both greater than 10.

so, given a level of significance of % , critical value = upper % of N(0,1), and

p-value = P(Z > Zcal)

Using Excel Functions : Statistical NORMSINV and NORMDIST , critical value and p-value are found to be as shown in the above table.

DECISION :

Since p-value > , Ho is accepted.

CONCLUSION :

There is not enough evidence to suggest that the claim is valid.

P = 0.55

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0.55 x 0.45 C.I = 0.55 +1.81 40

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npo 1 - Po)

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