Answer:
(a) 35 heads correspond to -3 on the standard scale.
(b) z = 2.4 corresponds to 62 heads on the number of heads scale.
Step-by-step explanation:
(a) If we flip a fair coin once, probability of getting head = 0.5
If we flip a fair coin 100 times, mean number of heads = 100(0.5) = 50
If there are N draws with a P probability of success, the standard deviation (SD) is given as:
[tex]SD = \sqrt{(N)(P)(1 - P)}[/tex]
Here, the probability of getting a head (P) is 0.5 while the number of draws (N) is 100. So,
[tex]SD = \sqrt{(100)(0.5)(1-0.5)}[/tex]
SD = 5
The standard scale value is: (35 - 50) / 5 = -3
Hence, 35 heads correspond to -3 on the standard scale.
(b) The standard scale value is 2.4 and we need to find the number of heads.
(X - 50) / 5 = 2.4
X - 50 = 12
X = 62
Hence, z = 2.4 on the standard scale corresponds to 62 on the number of heads scale.
The 35 heads corresponds to -3 on the standard scale, and the number of heads is 62
The head on the standard scale
The given parameter is:
n = 100
In a flip of a coin, the probability of a head is:
p = 1/2
So, the mean of the distribution is:
[tex]\bar x = np[/tex]
[tex]\bar x = 100 * 1/2[/tex]
[tex]\bar x = 50[/tex]
The standard deviation is:
[tex]\sigma = \sqrt{np(1 - p)}[/tex]
So, we have:
[tex]\sigma = \sqrt{100 * 1/2 * (1 - 1/2)}[/tex]
[tex]\sigma = 5[/tex]
The corresponding value on the standard scale is then calculated as:
[tex]z= \frac{x - \mu}{\sigma}[/tex]
For 35 heads, we have:
[tex]z= \frac{35 - 50}{5}[/tex]
[tex]z = -3[/tex]
Hence, the 35 heads corresponds to -3 on the standard scale
(b) The number of heads
We have:
[tex]z= \frac{x - \mu}{\sigma}[/tex]
When z = 2.4, the equation becomes
[tex]2.4= \frac{x - 50}{5}[/tex]
Multiply both sides by 5
[tex]12= x - 50[/tex]
Add 50 to both sides
[tex]x = 62[/tex]
Hence, the number of heads is 62
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