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Sampling Distribution of Sample Proportion

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1) A store has determined that 25% of all sales are credit sales. A random sample of 75 sales is selected. What is the probability that the sample proportion will be:
a) greater than .34?
b) between .196 and .354?
c) less than .25?
d) less than .10?

2) 10% of items produced by a machine are defective. A random sample of 100 items is selected and checked for defects. What is the probability that the sample will contain:
a) more than .025?
b) less than .13 defective units?

https://brainmass.com/statistics/probability/sampling-distribution-sample-proportion-406061

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Probability questions/need answers, but explanation of solution
1) A store has determined that 25% of all sales are credit sales. A random sample of 75 sales is selected. What is the probability that the sample proportion will be:
Let X be the proportion of credit sales. Then X can be assumed to be normal with mean, Âµ = 0.25 and standard deviation Ïƒ = = = 0.05. Standardizing the variable X using and from standard normal tables, we can find the probabilities as follows.
a) Greater than .34?
P (X > 0.34) = = P (Z > 1.8) = 0.0359

b) Between .196 and .354?
P (0.196 < X < 0.354) =
= P (-1.08 < Z < 2.08)
= 0.8412

c) Less than .25?
P (X < 0.25) = = P (Z < 0) = 0.5000

d) Less than .10?
P (X < 0.10) = = P (Z < -3) = 0.00135

2) 10% of items produced by a machine are defective. A random sample of 100 items is selected and checked for defects. What is the probability that the sample will contain:
Let X be the defective items produced by the machine. Then X can be assumed to be normal with mean, Âµ = 0.10 and standard deviation Ïƒ = = = 0.03. Standardizing the variable X using and from standard normal tables, we can find the probabilities as follows.
a) More than .025?
P (X > 0.025) = = P (Z > -2.5) = 0.9938

b) Less than .13 defective units?
P (X < 0.13) = = P (Z < 1) = 0.8413

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