The lenghts of botls produced in a factory may be taken to be normally distribut

Question

The lenghts of botls produced in a factory may be taken to be normally distributed. The bolts are checked on two "go-no go" gauges and those shorter than 2.9 or longer than 3.1 inches are rejected. A random sample of 397 bolts are checked on the gauges. If the (true but unknown) mean length of the bolts produced at the taime was 3.06 inches and the (true but unknown) standard deviation was .03 inches, what values would you expect for the number of bolts found to be too short and the number of bolts found to be too long? And a random sample of 50 bolts from another factory are also checked. If for these the number of bolts too short and too long are 12 each, estimate the mean and standard deviation for these bolts. State any assumptions.

Explanation / Answer

1)as from normal distribution Z =(X-mean)/std deviation

for random sample of 397 bolts:

P(X<2.9) =P(Z<(2.9-3.06)/0.03) =P(Z<(-5.3333) =0.0000

hence number of short bolts =np =397*0 =0

P(Z>3.1) =1-P(Z<(3.1-3.06)/0.03) =1-P(Z<1.333) =1-0.9088 =0.0912

hence number of longer bolts =np=397*0.0912 =36 (by rounding up)

for a random sample of 50.

let mean =x and std deviaiton =y

number of short bolts =12

hence probabilty =12/50 =0.24

for which z value for smaller bots =-0.595

and z value for longer bolts =0.595

hence x-0.595y =2.9

and x+0.595y =3.1

therefore from above x =3.0 =mean

and y =0.168 =std deviation

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.