Ouestion 1 (i) Tom a statistician surveyed NZ women\'s height. He surveyed each

Question

Ouestion 1 (i) Tom a statistician surveyed NZ women's height. He surveyed each 55 women and found that X = 178.4 cm. He also learned from Statistics NZ that the population of NZ women has = 180.0 cm, with = 10.4cm. a) Calculate the standard error of the mean ( b) In random samples of 55 women, what is the probability of obtaining a sample mean X> 178.4 cm c) Why does the standard error of the mean decrease as the sample size n increases from 55 to 60 women? (ii) Suppose that the weights of fertilizer bags X from a production line have a Normal Distribution with = 20 (kilos) and = 1 (kilos). The firm wants to know how many fertilizer bags contain a particular bacteria. The firm believes this is relatively rare and that the population proportion is 2%. A random sample of 28 bags is tested and the sample proportion with the bacteria is found to be 5%. Would it be appropriate to apply the Central Limit Theorem in this case? Explain your reasoning.

Explanation / Answer

SolutionA:

mean=180

sd=10.4

n=55

standard error of mean=sd/sqrt(samplesize)

=10.4/sqrt(55)

SE =1.402336

Solutionb:

z=x-mean/sd/sqrt(n)

P(X>178.4)

n=55

z=x-mean/sd/sqrt(n)

=178.4-180/10.4/sqrt(55)

= -1.140953

P(Z>-1.141)

P(z<1.141)=0.8729

ANSWER:0.8729

RCODE FOR SAME IS

pnorm(178.4,mean=180, sd=1.402336, lower.tail=FALSE)

OUTPUT:

0.8730553

Solutionc:

sample size=6

standard error of mean=std dev/sqrt(n)

=10.4/sqrt(60)

SE =1.34264

As sample size increases from 55 to 60 standard error of mean increases.   

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