Respuesta :
Answer:
(a) P (42 ≤ X ≤ 58) = 0.9108
(b) P (X ≥ 50) = 0.5398
Step-by-step explanation:
The random variable X is defined as the number of heads.
The probability of tossing a Heads in a single flip is, P (X) = p = 0.50.
The coin was flipped n = 100 times.
The random variable thus follows a Binomial distribution with parameters n and p.
As the sample size is too large (n > 30) and the probability of success is closer to 0.50, the binomial distribution can be approximated by the Normal distribution.
The mean of this distribution is: [tex]\mu=np=100\times0.50=50[/tex].
The standard deviation of this distribution is:[tex]\sigma=\sqrt{np(1-p)}=\sqrt{100\times0.50\times(1-0.50)}=5[/tex]
(a)
Compute the value of P (42 ≤ X ≤ 58) by applying the continuity correction as follows:
After applying the continuity correction the probability statement is:
[tex]P (42-0.5 \leq X \leq 58+0.5)=P(41.5< X< 58.5)[/tex]
The probability is:
[tex]P(41.5<X< 58.5)=P(\frac{41.5-50}{5}< X< \frac{58.5-50}{5})\\=P(-1.75<Z<1.7)\\=P(Z<1.7)-P(Z<-1.7)\\=P(Z<1.7)-[1-P(Z<1.7)]\\=2P(Z<1.7)-1\\=(2\times0.9554)-1\\=0.9108[/tex]
*Use a z-table for the probability value.
Thus, the value of P (42 ≤ X ≤ 58) by applying the continuity correction is 0.9108.
(b)
Compute the value of P (X ≥ 50) by applying the continuity correction as follows:
After applying the continuity correction the probability statement is:
[tex]P(X\geq 50-0.5)=P(X>49.5)[/tex]
The probability is:
[tex]P(X>49.5)=P(\frac{X-\mu}{\sigma}>\frac{49.5-50}{5} )\\=P(Z>-0.1)\\=P(Z<0.1)\\=0.5398[/tex]
Thus, the value of P (X ≥ 50) by applying the continuity correction is 0.5398.
