Answer: 0.5467
Step-by-step explanation:
We assume that the test scores for adults are normally distributed with
Mean : [tex]\mu=100[/tex]
Standard deviation : [tex]\sigma=20[/tex]
Sample size : = 50
Let x be the random variable that represents the IQ test scores for adults.
Z-score : [tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x =85
[tex]z=\dfrac{85-100}{20}\approx-0.75[/tex]
For x =115
[tex]z=\dfrac{115-100}{20}\approx0.75[/tex]
By using standard normal distribution table , the probability the mean of the sample is between 95 and 105 :-
[tex]P(85<X<115)=P(-0.75<z<0.75)=1-2(P(z<-0.75))\\\\=1-2(0.2266274)=0.5467452\approx0.5467[/tex]
Hence, the probability that a randomly selected adult has an IQ between 85 and 115 =0.5467
