Sports: Surfing In Hawaii, January is a favorite month for surfing since 60% of the days have a surf of at least 6 feet (Reference: Hawaii Data Book, Robert C. Schmitt). You work day shifts in a Honolulu hospital emergency room. At the beginning of each month you select your days off, and you pick 7 days at random in January to go surfing. Let r be the number of days the surf is at least 6 feet.
(a) Make a histogram of the
(b) What is the probability of getting 5 or more days when the surf is at least 6 feet?
(c) What is the probability of getting fewer than 3 days when the surf is at least 6 feet?
(d) What is the expected number of days when the surf will be at least 6 feet?
(e) What is the standard deviation of the r-probability distribution?
(d) Interpretation Can you be fairly confident that the surf will be at least 6 feet high on one of your days off? Explain.
(a)
To graph: The histogram.
Explanation of Solution
Given: 60% of the days in January have a surf height of at least 6 feet and the number of days randomly picked in January is 7.
Graph:
According to the provided details, the number of days the surf is at least 6 feet high, r follows the binomial distribution with the probability of success in a single trial (p) is 0.60 and the number of trials (n) are 7.
Consider, the probability values provided in Table 2 of the appendix for
The probability values for
Follow the steps given below to obtain the histogram:
Step 1: Place the values of r- 0, 1, 2, 3, 4, 5, 6, and 7 on the horizontal axis.
Step 2: Place the values of
Step 3: Construct a bar over each of the r values (0, 1, 2, 3, 4, 5, 6, 7) ranging from
The histogram for the binomial distribution for
Interpretation: The graph above displays the fact that the binomial distribution having
(b)
To find: The probability of getting 5 or more days when the surf is at least 6 feet high.
Answer to Problem 11P
Solution: The probability is 0.420.
Explanation of Solution
Given: The provided values are:
Calculation: The random variable ‘r’ follows the binomial distribution with the parameters,
The probability of getting 5 or more days when the surf is at least 6 feet high can be calculated by:
Consider, the probability values provided in the Table 2 of the appendix for
The probability values for
Substitute the values in the above formula,
The required probability is 0.420.
Interpretation: There is a 42% chance of getting 5 or more days when the surf is at least 6 feet high.
(c)
To find: The probability of getting fewer than 3 days when the surf is at least 6 feet high.
Answer to Problem 11P
Solution: The probability is 0.096.
Explanation of Solution
Given: The provided values are:
Calculation: The random variable ‘r’ follows the binomial distribution with the parameters
The probability of getting fewer than 3 days when the surf is at least 6 feet high can be calculated by:
Consider, the probability values provided in the Table 2 of the appendix for
The probability values for
Substitute the values in the above formula. Thus,
The required probability is 0.096.
Interpretation: There is a 9.6% chance of getting fewer than 3 days when the surf is at least 6 feet high.
(d)
To find: The expected number of days when the surf will be at least 6 feet high.
Answer to Problem 11P
Solution: The expected value is 4.2.
Explanation of Solution
Given: The provided values are:
Calculation: The random variable ‘r’ follows the binomial distribution with the parameters,
The formula that is used to calculate the expected value of the binomial distribution is:
Substitute the provided values in the above formula,
The expected value is 4.2.
Interpretation: One can expect 4.2 out of 7 days when the surf will be at least 6 feet high.
(e)
To find: The standard deviation of the r-distribution.
Answer to Problem 11P
Solution: The standard deviation is 1.296.
Explanation of Solution
Given: The provided values are:
Calculation: The random variable ‘r’ follows the binomial distribution with the parameters,
The formula that is used to calculate the standard deviation of the binomial distribution is:
Substitute the provided values in the above formula,
The standard deviation is 1.296.
Interpretation: The standard deviation of the probability distribution of r is 1.296.
(f)
To explain: Whether one can be fairly confident that the surf will be at least 6 feet high on one of your days off.
Answer to Problem 11P
Solution: Yes, one can be fairly confident as the expected number of days the surf will be at least 6 feet high is 4. The probability of getting at least 1 day out of 7 during which the surf will be at least 6 feet high is 0.998.
Explanation of Solution
Given: The provided values are:
Calculation: The random variable ‘r’ follows the binomial distribution with the parameters,
The probability of getting at least 1 day out of 7 during which the surf will be at least 6 feet high can be calculated by:
Consider, the probability values provided in Table 2 of the appendix for
The probability value for
Substitute the values in the above formula. Thus,
The probability is 0.998.
Interpretation: The expected number of days that the surf will be at least 6 feet high is approximately 4 and there is a 99.8% chance of getting a surf that is at least 6 feet high on one of the days off out of 7 days. So, one can be confident of getting it.
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Chapter 6 Solutions
Understanding Basic Statistics
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