Question

Can someone please help me with my central limit theorem homework? Thank you much!

1. A population has parameters μ=36.3 and 57.1. You intend to draw a random sample of size n=139.

a. What is the mean of the distribution of sample means?
μ¯x= _________________

b. What is the standard deviation of the distribution of sample means?
(Report answer accurate to 2 decimal places.)

σx¯=_______________

2. A population of values has a normal distribution with μ=201.8μ=201.8 and σ=90.9σ=90.9. You intend to draw a random sample of size n=20.

a. Find the probability that a single randomly selected value is less than 266.8.
P(X < 266.8) = _________

b. Find the probability that a sample of size n=20 is randomly selected with a mean less than 266.8.
P(M < 266.8) = __________

Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

3. A manufacturer knows that their items have a lengths that are skewed right, with a mean of 16.1 inches, and standard deviation of 3.1 inches.
If 45 items are chosen at random, what is the probability that their mean length is greater than 17.4 inches?

(Round answer to four decimal places)

4. A particular fruit's weights are normally distributed, with a mean of 226 grams and a standard deviation of 11 grams.
If you pick 14 fruits at random, then 2% of the time, their mean weight will be greater than how many grams?

Give your answer to the nearest gram.

Answer 1

1)

a)

$\mu$$\bar{x}$ = $\mu$ = 36.5

b)

$\sigma$$\bar{x}$ = $\sigma$ / sqrt(n)

= 57.1 / sqrt(139)

= 4.84

2)

a)

Given, $\mu$ = 201.8 , $\sigma$ = 90.9

We convert this to standard normal as

P(X < x) = P(Z < x -  $\mu$ / $\sigma$ )

So,

P(X < 266.8) = P(Z < (266.8 - 201.8) / 90.9 )

= P(Z < 0.72)

= 0.7642

b)

Using central limit theorem,

P($\bar{x}$ < x) = P(Z < (x -  $\mu$) / ($\sigma$ / sqrt(n) ) )

So,

P( $\bar{x}$ < 266.8) = P(Z < (266.8 - 201.8) / (90.9 / sqrt(20) ) )

= P(Z < 3.20)

= 0.9993​​​​​​​