Answered By Sara S

$\mu$ $$\mu$$

Answered By Sara S

Hi there, 

Bottle Caps :

We use z-score method. 

Let  $\mu$μ  = mean = 2.8 and  $\sigma$σ = standard deviation = 0.03.

Then, we have z =  $\frac{X-\mu}{\sigma}$X−‍μσ . Accordingly, we have 

P((2.74- $\mu$μ )/ $\sigma$σ  $\le$≤  z  $\le$≤ (2.86- $\mu$μ )/ $\sigma$σ ) = P((2.74 - 2.8)/0.03  $\le$≤ z $\le$≤ (2.86 - 2.8)/0.03)=

P(-0.06/0.03   $\le$≤   z  $\le$≤ 0.06/0.03)= P(-2  $\le$≤ z  $\le$≤ 2) = P(z  $\le$≤ 2) - P(z $\le$≤ -2). 

Form the z-table, we have 

P(z $\le$≤ 2 )= 0.9772 and P(z $\le$≤ -2) = 0.0228. 

Therefore, we have 

P(2.74  $\le$≤  X  $\le$≤ 2.86) = 0.9772 - 0.0228 = 0.9544

For all other parts, use the same method and let me know if you have any questions. 

Answered By Sara S

Correction: 

In the second line, z =  $\frac{X-\mu}{\sigma}$X−μσ  

Answered By Sara S

Men's Shoe Designer:

Based on the 95% Rules, we have:

Movin 2 standard deviations to right of the mean and also to the left of the mean, we cover 95% of values. In the other word, 

P( $\mu-2\sigma\le$μ−2σ≤ X $\le\mu+2\sigma$≤μ+2σ )  $\approx$≈ 95%

Since the guy is focusing on 95% of the Male population, we conclude:

 the smallest foot length the designed is willing to accommodate = $\mu-2\sigma$μ−2σ  = 25.25 - 2(3.875) = 17.5

and

 the largest foot length the designed is willing to accommodate  $\approx$≈  $\mu+2\sigma$μ+2σ  = 25.25+2(3.875) = 33

Hope this helps!

Good Day!