Question

A statistical analysis of​ 1,000 long-distance telephone calls made by a company indicates that the length of these calls is normally​ distributed, with a mean of

280280

seconds and a standard deviation of

3030

seconds. Complete parts​ (a) through​ (d).

a.

What is the probability that a call lasted less than

230230

​seconds?The probability that a call lasted less than

230230

seconds is

. 0478.0478 .

​(Round to four decimal places as​ needed.)

b.

What is the probability that a call lasted between

230230

and

340340

​seconds?The probability that a call lasted between

230230

and

340340

seconds is

. 9295.9295 .

​(Round to four decimal places as​ needed.)

c.

What is the probability that a call lasted more than

340340

​seconds?The probability that a call lasted more than

340340

seconds

. 9772.9772 .

​(Round to four decimal places as​ needed.)

d.

What is the length of a call if only

2.5 %2.5%

of all calls are​ shorter?

2.52.5​%

of the calls are shorter than

nothing

seconds.

​(Round to two decimal places as​ needed.)

A statistical analysis of​ 1,000 long-distance telephone calls made by a company indicates that the length of these calls is normally​ distributed, with a mean of

280280

seconds and a standard deviation of

3030

seconds. Complete parts​ (a) through​ (d).

a.

What is the probability that a call lasted less than

230230

​seconds?The probability that a call lasted less than

230230

seconds is

. 0478.0478 .

​(Round to four decimal places as​ needed.)

b.

What is the probability that a call lasted between

230230

and

340340

​seconds?The probability that a call lasted between

230230

and

340340

seconds is

. 9295.9295 .

​(Round to four decimal places as​ needed.)

c.

What is the probability that a call lasted more than

340340

​seconds?The probability that a call lasted more than

340340

seconds

. 9772.9772 .

​(Round to four decimal places as​ needed.)

d.

What is the length of a call if only

2.5 %2.5%

of all calls are​ shorter?

2.52.5​%

of the calls are shorter than

nothing

seconds.

​(Round to two decimal places as​ needed.)

Answer 1

Given, .

= 280, = 30

We convert this to standard normal as

P( X < x) = P( Z < x - / )

a)

P( X < 230) = P( Z < 230 - 280 / 30)

= P( Z < -1.6667)

= 0.0478

b)

P( 230 < X < 340) = P( X < 340) - P( X < 230)

= P( Z < 340 - 280 / 30) - P( Z < 230 - 280 / 30)

= P( Z < 2) - P( Z < -1.6667)

= 0.9772 - 0.0478

= 0.9295

c)

P( X > 340) = P( Z > 380 - 340 / 30)

= P( Z > 1.3333) .

= 0.0912

d)

We have to calculate x such that

P( X < x) = 0.025

That is

P( Z < x - / ) = 0.025

From Z table, z-score for the probability of 0.025 is -1.96

x - / = -1.96

x - 280 / 30 = -1.96

x = 221.2