SOLUTION: A manufacturer of submersible pumps claims that at most 30% of the pumps require repairs within the first 5 years of operation. If a random sample of 120 of these pumps includes 4

Question 894052: A manufacturer of submersible pumps claims that at most 30% of the pumps require repairs within the first 5 years of operation. If a random sample of 120 of these pumps includes 47 which required repairs within the first 5 years, test the null hypothesis p = 0.30 against the alternative hypothesis p> 0.30 at the 0.05 level of significance.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
this is a one tailed test because the alternate hypothesis is greater than rather than not equal to.

the population mean is estimated from the sample as n*p = 120*.3 = 36

the sample mean is 47.

the standard error is equal to sqrt (n * p * q) which is equal to sqrt(120 * .3 * .7) which is equal to sqrt(25.2) which is equal to 5.01996 rounded to 5 decimal places.

at .05 significance level, the critical z factor is 1.645.

if the z factor of the sample is greater than that, then the results are statistically significant and you can reject the null hypotheses which states that the average number of defective pumps is less than or equal to 36.

the results are statistically significant because the z factor of the sample is (47 - 36) / 5.01996 = 2.19.

the probability of getting a z score of 2.19 or greater is equal to .014 which is significantly less than .05.

you could also solve this problem as a ratio and you will get the same answer.

the population mean is .3

the sample mean is 47 / 120 = .391667.

the standard deviation is sqrt(p*q/n) which is equal to sqrt(.3*.7/120) which is equal to .041833.

the z score for this is (.391667 - .3) / .041833 which is equal to 2.19...

you get the same z score therefore the same result.

when you work the ratio problem as a mean, your formulas are:

m = n*p
s = sqrt(n*p*q)
p = probability of occurrence.
q = 1 - p

when you work the ratio problem as a ratio, your formulas are:

m = p
s = sqrt(p*q/n)
p = probability of occurrence.
q = 1 - p

the formula related because:

n*p / n = p

sqrt(n*p*q) / n = sqrt(n*p*q/n^2) = sqrt(p*q/n)

s is equal to the standard error of the distribution of sample means in this case.

s sometimes represents the standard deviation of the population.
s sometimes represents the standard deviation of the sample.
s sometime represents the standard deviation of the distribution of sample means.

in this case, s means the latter.
it's always good to tell peoople what s represents if you are using s for any of these 3 options.

a normal distribution is assumed.

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