Question

The mean incubation time of fertilized eggs is 21 days. Suppose the incubation times are approximately normally distributed with a standard deviation of 1 day.(a) Determine the 20th percentile for incubation times.(b) Determine the incubation times that make up the middle 39% of fertilized eggs.

Answer 1

Answer:

a) 20.16; b) 20.49 and 21.51

Step-by-step explanation:

We use z scores for each of these.  The formula for a z score is

$z=\frac{X-\mu}{\sigma}$.

For part a, we want the 20th percentile; this means we want 20% of the data to be lower than this. We find the value in the cells of the z table that are the closest to 0.20 as we can get; this is 0.2005, which corresponds with a z score of -0.84.

Using this, 21 as the mean and 1 as the standard deviation,

-0.84 = (X-21)/1

-0.84 = X-21

Add 21 to each side:

-0.84+21 = X-21+21

20.16 = X

For part b, we want the middle 39%.  This means we want 39/2 = 19.5% above the mean and 19.5% below the mean; this means we want

50-19.5 = 30.5% = 0.305 and

50+19.5 = 69.5% = 0.695.

Looking these values up in the cells of the z table, we find those exact values; 0.305 corresponds with z = -0.51 and 0.695 corresponds with z = 0.51:

-0.51 = (X-21)/1

-0.51 = X-21

Add 21 to each side:

-0.51+21 = X-21+21

20.49 = X

0.51 = (X-21)/1

0.51 = X-21

Add 21 to each side:

0.51+21 = X-21+21

21.51 = X

Answer 2

The required incubation times for the 20th percentile and middle 39 percentile are 20.16 and (between 20.49 and 21.51) respectively.

Given the Parameters :

• Mean, μ = 21
• Standard deviation, σ = 1

Recall the Zscore formula:

• Zscore = (X - μ) ÷ σ

The 20th percentile :

Using the normal distribution table ; P(Z < 0.20) has a Zscore of - 0.84

-0.84 = (X - 21) / 1

-0.84 = X - 21

X = - 0.84 + 21

X = 20.16

B.) The time which makes up the middle 39%

• Middle 39% = 39%/2 = 19.5% below and above the mean
• Lower = mean - 19.5% = 50% - 19.5% = 30.5%
• Upper = mean + 19.5% = 50% + 19.5% = 69.5%

Using the normal distribution table :

Zscore of the lower and upper percentile equals - 0.51 and 0.51 respectively

Using the Zscore formula :

Lower score :

-0.51 = (X - 21) / 1

-0.51 = X - 21

X = - 0.51 + 21

X = 20.49

Upper score :

0.51 = (X - 21) / 1

0.51 = X - 21

X = 0.51 + 21

X = 21.51

Therefore, the incubation times are 20.16 and between 20.49 to 21.51

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