Question

1.)

A particular fruit's weights are normally distributed, with a mean of 601 grams and a standard deviation of 34 grams.

If you pick 2 fruit at random, what is the probability that their mean weight will be between 599 grams and 668 grams

2.)

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 225.1-cm and a standard deviation of 1.4-cm. For shipment, 15 steel rods are bundled together.

Find the probability that the average length of the rods in a randomly selected bundle is between 224-cm and 225.6-cm.


P(224-cm <X¯ < 225.6-cm) = Round to 4 decimal places.

3.)

The manager of a computer retails store is concerned that his suppliers have been giving him laptop computers with lower than average quality. His research shows that replacement times for the model laptop of concern are normally distributed with a mean of 3.5 years and a standard deviation of 0.4 years. He then randomly selects records on 54 laptops sold in the past and finds that the mean replacement time is 3.4 years.

Assuming that the laptop replacement times have a mean of 3.5 years and a standard deviation of 0.4 years, find the probability that 54 randomly selected laptops will have a mean replacement time of 3.4 years or less.

P(¯¯¯X≤3.4 years)P(X¯≤3.4 years) = Round to 4 decimal places.

Based on the result above, does it appear that the computer store has been given laptops of lower than average quality?

• Yes. The probability of obtaining this data is less than 5%, so it is unlikely to have occurred by chance alone.
• No. The probability of obtaining this data is greater than 5%, high enough to have been a chance occurrence.

4.)

The amounts of nicotine in a certain brand of cigarette are normally distributed with a mean of 0.883 g and a standard deviation of 0.281 g. The company that produces these cigarettes claims that it has now reduced the amount of nicotine. The supporting evidence consists of a sample of 32 cigarettes with a mean nicotine amount of 0.828 g.

Assuming that the given mean and standard deviation have NOT changed, find the probability of randomly selecting 32 cigarettes with a mean of 0.828 g or less.


P(X¯ < 0.828 g) =  Round to 4 decimal places.

Answer 1

Question 1

$X \sim N ( \mu = 601 , \sigma = 34 )$
P ( 599 < X < 668 )
Standardizing the value
$Z = ( X - \mu ) / (\sigma/\sqrt{n})$
$Z = ( 599 - 601 ) / ( 34 /\sqrt{ 2 })$
Z = -0.08
$Z = ( 668 - 601 ) / ( 34 /\sqrt{ 2 })$
Z = 2.79
P ( -0.08 < Z < 2.79 )
P ( 599 < X < 668 ) = P ( Z < 2.79 ) - P ( Z < -0.08 )
P ( 599 < X < 668 ) = 0.9973 - 0.4669
P ( 599 < X < 668 ) = 0.5305

Question 2

$X \sim N ( \mu = 225.1 , \sigma = 1.4 )$
P ( 224 < X < 225.6 )
Standardizing the value
$Z = ( X - \mu ) / (\sigma/\sqrt{n})$
$Z = ( 224 - 225.1 ) / ( 1.4 /\sqrt{ 15 })$
Z = -3.04
$Z = ( 225.6 - 225.1 ) / ( 1.4 /\sqrt{ 15 })$
Z = 1.38
P ( -3.04 < Z < 1.38 )
P ( 224 < X < 225.6 ) = P ( Z < 1.38 ) - P ( Z < -3.04 )
P ( 224 < X < 225.6 ) = 0.9167 - 0.0012
P ( 224 < X < 225.6 ) = 0.9155