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Stella [2.4K]
3 years ago
6

A fertilizer company manufactures 10-pound bags of fertilizer with a standard deviation of 0.24 pounds per bag. The bag weights

are normally distributed. What is the probability that a sample of 4 bags will have a mean weight less than 9.8 pounds
Mathematics
1 answer:
Ksju [112]3 years ago
3 0

Answer:

the probability that a sample of 4 bags will have a mean weight less than 9.8 pounds is 0.05

Step-by-step explanation:

Given the data in the question;

μ_x = 10 pound bags

standard deviation s_x = 0.24 pounds

sample size n = 4

The bag weights are normally distributed so;

p( x' less than 9.8 ) will be;

p(  (x'-μ_x' / s_x')   <   (9.8-μ_x' / s_x')  )

we know that;

μ_x' = μ_x = 10

and s_x' = s_x/√n = 0.24/√4

so; we substitute

p(  z  <  ( (9.8 - 10) / (0.24/√4)  )

p(  z  <  -0.2 / 0.12   )

p(  z  <  -1.67   )

{ From z-table }

⇒ p(  z  <  -1.67   ) = 0.0475 ≈ 0.05

Therefore, the probability that a sample of 4 bags will have a mean weight less than 9.8 pounds is 0.05

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Answer:

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For this case we know this:

n=37 ,  p=0.2

We can find the standard error like this:

SE = \sqrt{\frac{\hat p (1-\hat p)}{n}}= \sqrt{\frac{0.2*0.8}{37}}= 0.0658

So then our random variable can be described as:

p \sim N(0.2, 0.0658)

Let's suppose that the question on this case is find the probability that the population proportion would be higher than 0.4:

P(p>0.4)

We can use the z score given by:

z = \frac{p -\mu_p}{SE_p}

And using this we got this:

P(p>0.4) = 1-P(z< \frac{0.4-0.2}{0.0658}) = 1-P(z

And we can find this probability using the Ti 84 on this way:

2nd> VARS> DISTR > normalcdf

And the code that we need to use for this case would be:

1-normalcdf(-1000, 3.04; 0;1)

Or equivalently we can use:

1-normalcdf(-1000, 0.4; 0.2;0.0658)

Step-by-step explanation:

We need to check if we can use the normal approximation:

np = 37 *0.2 = 7.4 \geq 5

n(1-p) = 37*0.8 = 29.6\geq 5

We assume independence on each event and a random sampling method so we can conclude that we can use the normal approximation and then ,the population proportion have the following distribution :

p \sim N(p,\sqrt{\frac{p(1-p)}{n}})

For this case we know this:

n=37 ,  p=0.2

We can find the standard error like this:

SE = \sqrt{\frac{\hat p (1-\hat p)}{n}}= \sqrt{\frac{0.2*0.8}{37}}= 0.0658

So then our random variable can be described as:

p \sim N(0.2, 0.0658)

Let's suppose that the question on this case is find the probability that the population proportion would be higher than 0.4:

P(p>0.4)

We can use the z score given by:

z = \frac{p -\mu_p}{SE_p}

And using this we got this:

P(p>0.4) = 1-P(z< \frac{0.4-0.2}{0.0658}) = 1-P(z

And we can find this probability using the Ti 84 on this way:

2nd> VARS> DISTR > normalcdf

And the code that we need to use for this case would be:

1-normalcdf(-1000, 3.04; 0;1)

Or equivalently we can use:

1-normalcdf(-1000, 0.4; 0.2;0.0658)

7 0
2 years ago
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