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A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24pounds, and the highest is 26pounds. Each weight is equally likely so the distribution of weights is uniform. A sample of 100weights is taken.

a. What is the distribution for the weights of one 25-pound lifting weight? What is the mean and standard deviation?

b. What is the distribution for the mean weight of 100,25-pound lifting weights?

c. Find the probability that the mean actual weight for the 100 weights is less than 24.9.

Short Answer

Expert verified

(a) The mean and standard deviation are 25and 0.577

(b) The distribution for the mean weight is X¯~N(25,.0577).

(c) The Probability is4.15%.

Step by step solution

01

Given Information Part(a)

According to the provided details, a manufacture produces 25-pound weights. The lowest weight is 24pounds and the highest weight is 26 pounds. Each weight is equally likely and the sample size considered 100 weights.

02

Explanation Part(a)

The distribution for weights of one 25pound lifting weight will follow the uniform distribution because the weights are equally likely. Thus, the uniform distribution of the lowest 24pounds and the highest 26pounds weight are given as:

X~U(a,b)

X~U(24,26)

The mean of the uniform distribution is given as:

μX=a+b2

=24+262

=25

And the standard deviation is given as:

localid="1648803309552" σx=(ba)212

localid="1648803315274" =(2624)212

=412

=0.577

03

Given Information Part(b)

According to the provided details, a manufacture produces 25-pound weights. The lowest weight is 24pounds and the highest weight is 26 pounds. Each weight is equally likely and the sample size considered 100 weights.

04

Explanation Part(b)

The distribution of mean weight of 100,25-pounds is given as:

X¯~NμX,σX/n

X¯~N(25,.5777/100)

X¯~N(25,.0577)

05

Given Information Part(c)

According to the provided details, a manufacture produces 25-pound weights. The lowest weight is 24pounds and the highest weight is 26 pounds. Each weight is equally likely and the sample size considered 100 weights.

06

Explanation Part(c)

To calculate the probability that P(ΣX<24.9), use Ti-83calculator. For this, click on 2nd, then DISTR and then scroll down to the normal CDF option and enter the provided details. After this, click on ENTER button of the calculator to have the desired result.

Therefore, the probability that the mean actual weight for the 100 weights is less than 24,9 is approximately4.15%

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