Question

A recent survey found that 63% of all adults over 50 wear glasses for driving. Ia random sample of 100 adults over 50 was selected., a). what is the expected value and standard deviation of the number of adults over 50 who wear glasses? b). Find the probability that more than 60 of the adults over 50 wear glasses. Hint: using normal distribution to approximate binomial distribution.

Answer 1

Here we have given that

p= population proportion of all adults over 50 who wear glasses for driving=63%=0.63

n=Number of adults over 50 = 100

Here, we need to find the probability that more than 60 of the adults over 50 wear glasses.

Now, n*p= 100*0.63 =63 > 10

n*p*(1-p)= 100*0.63*(1-0.63) = 23 >10

i.e. the normal distribution we can use to approximate to the binomial distribution.

(a)

We can find the expected value of the number of adults over 50 who wear glasses,

We know that the mean of the sample proportion is the population proportion.

= the expected value of the number of adults over 50 who wear glasses

=p

= 0.63

The expected value is 0.63

And

We can find the standard deviation of the number of adults over 50 who wear glasses

=   

=

= 0.0483

The standard deviation is 0.0483

Now, we need to find the probability that more than 60 of the adults over 50 wear glasses.

i.e =?

=

= 1-

= 1-

= 1-

=1 - 0.26763

=0.7324

the probability that more than 60 of the adults over 50 wear glasses is 0.7324

i.e =0.7324