Mathematics, 20.04.2020 20:25 shanedawson19

The weight of adobe bricks for construction is normally distributed with a mean of 3 pounds and a standard deviation of 0.25 pound. Assume that the weights of the bricks are independent and that a random sample of 25 bricks is chosen.(a) What is the probability that the mean weight of the sample is less than 3.05 pounds?
Round your answer to four decimal places (e. g. 98.7654).
(b) What value will the mean weight exceed with probability 0.99?
Round your answer to two decimal places (e. g. 98.76).

Answers: 2

Show answers

Answers

Answer from: freemanjskg

(a) The probability that the mean weight of the sample is less than 3.05 pounds is 0.8413.

(b) The value for which the mean weight exceed with probability 0.99 is 2.88 pounds.

Step-by-step explanation:

We are given that the weight of adobe bricks for construction is normally distributed with a mean of 3 pounds and a standard deviation of 0.25 pound.

Assume that the weights of the bricks are independent and that a random sample of 25 bricks is chosen.

Let $\bar X$ = sample mean weight of the bricks

The z-score probability distribution for sample mean is given by;

Z = $\frac{ \bar X-\mu}{\frac{\sigma}{\sqrt{n} } }} }$ ~ N(0,1)

where, $\mu$ = population mean weight of bricks = 3 pounds

$\sigma$ = standard deviation = 0.25 pounds

n = sample of bricks = 25

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

(a) Probability that the mean weight of the sample is less than 3.05 pounds is given by = P( $\bar X$ < 3.05 pounds)

P( $\bar X$ < 3.05 pounds) = P( $\frac{ \bar X-\mu}{\frac{\sigma}{\sqrt{n} } }} }$ < $\frac{3.05-3}{\frac{0.25}{\sqrt{25} } }} }$ ) = P(Z < 1) = 0.8413

So, in the z table the P(Z $\leq$ x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 1 in the z table which has an area of 0.84134.

Hence, the probability that the mean weight of the sample is less than 3.05 pounds is 0.8413.

(b) Now, we have to find the value for which the mean weight exceed with probability 0.99, that means;

P( $\bar X$ > x) = 0.99 {where x is the required value}

P( $\frac{ \bar X-\mu}{\frac{\sigma}{\sqrt{n} } }} }$ > $\frac{x-3}{\frac{0.25}{\sqrt{25} } }} }$ ) = 0.99

P(Z > $\frac{x-3}{\frac{0.25}{\sqrt{25} } }} }$ ) = 0.99

1 - P(Z $\leq$ $\frac{x-3}{\frac{0.25}{\sqrt{25} } }} }$ ) = 0.99

P(Z $\leq$ $\frac{x-3}{\frac{0.25}{\sqrt{25} } }} }$ ) = 1 - 0.99

P(Z $\leq$ $\frac{x-3}{\frac{0.25}{\sqrt{25} } }} }$ ) = 0.01

Now, in the z table, the critical value of x for which the probability area is less than 0.01 is -2.3263, i.e.;

$\frac{x-3}{\frac{0.25}{\sqrt{25} } }} }$ = -2.3263

${x-3}= -2.3263 \times \frac{0.25 }{\sqrt{25}}$

= 3 - 0.116315 = 2.88 pounds

Therefore, the value for which the mean weight exceed with probability 0.99 is 2.88 pounds.

Answer from: Quest

step-by-step explanation:

Answer from: Quest

1e+27

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 18:00

Julius sold five times as many computers as sam sold last year. in total, they sold 78 computers. how many computers did julius sell?

Answers: 1

Answer

Mathematics, 21.06.2019 19:00

Abarbecue sold 26 hamburgers and hot dogs. hamburgers sell for $3.50 each and hot dogs sell for $2.00 each. if the barbecue made $70 in sales, determine and state the number of hot dogs sold.

Answers: 1

Answer

Mathematics, 21.06.2019 22:00

Luke wants to reduce the area of his rectangular garden by 1/4 the expression 14/w can be used to represent this change. what is another way to write this expression?

Answers: 1

Answer

Mathematics, 21.06.2019 23:00

Graph the system of equations on your graph paper to answer the question. {y=−x+4y=x−2 what is the solution for the system of equations? enter your answer in the boxes.

Answers: 1

Answer

You know the right answer?

The weight of adobe bricks for construction is normally distributed with a mean of 3 pounds and a st...

Questions