The average supermarket lemon weighs about 100 grams. Let\'s assume = 100 and =

Question

The average supermarket lemon weighs about 100 grams. Let's assume = 100 and = 17.

a) Calculate Pr{Y < 93}

b) Now assume you take a sample of 15 lemons. Calculate Pr{ Y < 93}.

c) Now assume you take a sample of n = 6 lemons. What is the probability of the sample average being within 10 grams of the population mean (100 g)? In other words, you need to figure out Pr{ - 10 < Y < + 10}

d) What is the probability of the sample average being within 10 grams of the population mean if the population mean is actually = 90g? Are you surprised? Why or why not? You should make sure you understand what happened here.

e) Now assume a sample of n = 17 lemons and repeat (a). What is the effect of sample size?

Explanation / Answer

Please keep the z tables ready , we know that

Z = (x-mean)/sd

Pr{Y < 93}

(93-100)/17 = -0.4117

P ( Z<0.4117 )=1P ( Z<0.4117 )=10.6591=0.3409

b)

for n = 15 , the formula becomes

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

(93-100)/(17/sqrt(15)) = -1.594

P ( Z<1.594 )=1P ( Z<1.594 )=10.9441=0.0559

c) Now assume you take a sample of n = 6 lemons. What is the probability of the sample average being within 10 grams of the population mean (100 g)? In other words, you need to figure out Pr{ - 10 < Y < + 10}

n = 6

and we need to find

90 < Y < 110

(90-100)/(17/sqrt(6)) < Y < (110-100)/(17/sqrt(6))

-1.44 <Y < 1.44

To find the probability of P (1.44<Z<1.44), we use the following formula:

P (1.44<Z<1.44 )=P ( Z<1.44 )P (Z<1.44 )

We see that P ( Z<1.44 )=0.9251.

P ( Z<1.44 ) can be found by using the following fomula.

P ( Z<a)=1P ( Z<a )

After substituting a=1.44 we have:

P ( Z<1.44)=1P ( Z<1.44 )

We see that P ( Z<1.44 )=0.9251 so,

P ( Z<1.44)=1P ( Z<1.44 )=10.9251=0.0749

At the end we have:

P (1.44<Z<1.44 )=0.8502

d) for mean - 90

n = 6

and we need to find

80 < Y < 100

(80-90)/(17/sqrt(6)) < Y < (100-90)/(17/sqrt(6))

-1.44 <Y < 1.44

To find the probability of P (1.44<Z<1.44), we use the following formula:

P (1.44<Z<1.44 )=P ( Z<1.44 )P (Z<1.44 )

We see that P ( Z<1.44 )=0.9251.

P ( Z<1.44 ) can be found by using the following fomula.

P ( Z<a)=1P ( Z<a )

After substituting a=1.44 we have:

P ( Z<1.44)=1P ( Z<1.44 )

We see that P ( Z<1.44 )=0.9251 so,

P ( Z<1.44)=1P ( Z<1.44 )=10.9251=0.0749

At the end we have:

P (1.44<Z<1.44 )=0.8502

Please note that we can answer only 4 subparts of a question at a time , as per the answering guidelines

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