Concept explainers
(a)
To find: The probability of the sample proportion of heads between 0.4 and 0.6 by using normal to the binomial approximate.
(a)
Answer to Problem 53UYK
Solution: The probability is 0.9857.
Explanation of Solution
Calculation: When a coin is tossed, there are two possible outcomes, “heads” or “tails.” The probability of the heads coming up in a coin toss is calculated as:
The average of the sample proportion is calculated as:
The standard deviation of the sample proportion is calculated as:
Hence, the average value and standard deviation are 0.5 and 0.0408, respectively. Now, calculate the z- score,
The lower bound is
Calculate the z- score for the upper bound
Now, Excel is used to calculate the left tailed areas between the z scores. Use the formula
Use the formula
The area between them is calculated by subtracting the above values as:
Hence, the probability is 0.9857.
(b)
To find: The probability of the sample proportion of heads between 0.45 and 0.55 by using normal to the binomial approximate.
(b)
Answer to Problem 53UYK
Solution: The probability is 0.7813.
Explanation of Solution
Calculation: When a coin is tossed, there are two possible outcomes, “heads” or “tails.”
The probability of the heads coming up in a coin toss is calculated as:
The average of the sample proportion is calculated as:
The standard deviation of the sample proportion is calculated as:
Hence, the average value and standard deviation are 0.5 and 0.0408 respectively. Now, calculate the z- score,
The lower bound is
Calculate the z- score for the upper bound
Now, Excel is used to calculate the left tailed areas between the z scores. Use the formula
Use the formula
The area between them is calculated by subtracting the above values as:
Hence, the probability is 0.7813
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Chapter 5 Solutions
Introduction to the Practice of Statistics
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