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

a) What is the distribution for the sum of the weights of 100 25-pound lifting weights?

b) Find P(Σx < 2,450).

Given in the question that

A manufacturer produces 25-pound lifting weights

The lowest actual weight is 24 pounds

Highest is 26 pounds

Explanation:

The uniform distribution for lowest $2400$pounds and the highest $2600$pounds the weight are given as;

localid="1649393005338" $X\mathit{-}U\mathit{(}a\mathit{,}b\mathit{)}$

$X-U(2400,2600)$

The mean of the uniform distribution is;

${\mu }_x=\frac{a+b}{2}$

$=\frac{2400+2600}{2}$

$=2500$

The standard deviation is;

${\sigma }_x=\sqrt{\frac{(b-a{)}^2}{12}}$

$=\sqrt{\frac{(2600-2400{)}^2}{12}}$

$=57.735$

The mean weight is;

$\sum X~N\left({\mu }_X,{\sigma }_X/\sqrt{n}\right)$

Given in the question that

A manufacturer produces $25$-pound lifting weights

The lowest actual weight is $24-$pounds

Highest is $26$pounds

The formula for calculating probability is;

$P\left(\sum X<2450\right)$

For the calculation we use TI-83 calculator

For this, click on $2^{\text{nd}}$, then DISTR,

Scroll down to the normalcdf option and enter the provided details.

After this, click on ENTER button of the calculator to have the desired result.

The result shown the calculator is given below.