Answer: a) 0.5636 b) 0.7881
Step-by-step explanation:
We assume that women’s heights are normally distributed .
Let x be the random variable that represents the shoe sizes.
Also, The population mean = [tex]\mu=\text{63.6 inches}[/tex] ; Standard deviation: [tex]\sigma=\text{2.5 inches}[/tex]
a) Formula for z:-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
Put x= 64, we get
[tex]z=\dfrac{64-63.6}{2.5}=0.16[/tex]
Now, the probability that the male shoe sizes are greater than 8 :-
[tex]P(z<0.16)=0.5635595\approx0.5636[/tex]
Hence, the probability that her height is less than 64 inches = 0.5636
b. Sample size : n= 25
Then , the formula for z :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
Put x= 64, we get
[tex]z=\dfrac{64-63.6}{\dfrac{2.5}{\sqrt{25}}}=0.8[/tex]
Then, the probability that they have a mean height less than 64 inches.:_
[tex]P(z<0.8)=0.7881446\approx0.7881[/tex]
Hence, the probability that they have a mean height less than 64 inches. =0.7881
