A company produces batteries. On average, 85% of all batteries produced are good. Each battery is tested before being dispatched, and the inspector correctly classifies the battery 90% of the time.
A. What percentage of the batteries will be “classified as good”?
B. What is the probability that a battery is defective given that it was classified as good?
2. A teacher-student committee consisting of 4 people is to be formed from 5 teachers and 20 students. Find the probability that the committee will consist of at least one student.
3. From past, a company knows that in cartons of bulbs, 90% contain no defective bulbs, 5% contain one defective bulb, 3% contain two defective bulbs, and 2% contain three defective bulbs. Find the mean and standard deviation for the number of defective bulbs.
4. A study found that 25% of car owners in Fiji had their cars washed professionally rather than do it themselves. If 18 car owners are randomly selected, find the probability that at most two people have their cars washed professionally.
5. The average age of a breed of dog is 19.4 years. If the distribution of their ages is normal and 20% of dogs are older than 22.8 years, find the standard deviation.
6. Suppose that the lifespan of laptops is normally distributed with a mean of 24.3 months and standard deviation of 2.6 months. If USP provides its 33 staff with a laptop, find the probability that the mean lifespan of these laptops will be less 23.8 months.
7. A Fiji Travel Data Center survey reported that Fijians stayed an average of 7.5 nights when they went on vacation around the country. For the purpose of the survey, the sample size was 1500. Assume the population standard deviation was 0.8; find a point estimate of the population mean. In addition, find the 95% confidence interval of the true mean.
8. Prekindergarten studies has proved to be very effective for children. To illustrate its effectiveness, a study was carried out in Fiji and it was found that 73% of prekindergarten children ages 3 to 5 whose mothers had a bachelor’s degree or higher were enrolled in center-based early childhood care and education programs. This could also be of other obvious reasons. How large a sample is needed to estimate the true proportion within 3 percentage points with 95% confidence? How large a sample is needed if you had no prior knowledge of the proportion?
9. Rao’s Finance claims that less than 50% of adults in Suva have a will. A will is a legal document that sets forth your wishes regarding the distribution of your property and the care of any minor children. If you die without a will, those wishes may not be carried out. To substantiate the claim, a random sample of 1000 adults showed that 450 of them have a will.
A. At the 5% significance level, can you conclude that the percentage of people who have a will is less than 50%?
B. What is the Type I error in part A? What is the probability of making this error?
C. What would your decision be in part A if the probability of making a Type I error were zero? Explain.
10. Rewa Delta Union Rugby CEO has become concerned about the slow pace of the rugby games played in the current union rugby, fearing that it will lower the spectator attendance. The CEO meets with the union rugby managers and referees and discusses guidelines for speeding up and making the games more interesting and lively. Before the meeting, the mean duration of the 15-sided rugby game time was 3 hours, 5 minutes, that is, 185 minutes. This includes all the breaks and injury times during the game. A random sample of 36 of the 15-sided rugby games after the meeting showed a mean of 179 minutes with a standard deviation of 12 minutes. Testing at the 1% significance level, can you conclude that the mean duration of 15-sided union rugby games has decreased after the meeting?
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Q 1 Good Batteries = 85 %
Not good Batteries = 100 - 85 % = 15 %
inspector correctly classifies the battery 90% of the time.
=> 100 - 90 % = 10 % is incorrect classification
Classified Good batteries out of 85 % good = (90/100) 85 % = 76.5 %
Classified not good Batteries out of 85 % good = (10/100) 85 % = 8.5 %
Classified not Good batteries out of 15 % not good = (90/100) 15 % = 13.5 %
Classified Good Batteries out of 15 % not good = (10/100) 15 % = 1.5 %
percentage of the batteries classified as good = 76.5 + 1.5 = 78 %
Classified good and actual good = 76.5 %
Classified Good and actual not good ( defective) = 1.5 %
probability that a battery is defective given that it was classified as good
= (1.5)/(1.5 + 76.5)
= 1.5/78
= 1/52
= 1.92 %
78 % of the batteries will be “classified as good"
probability that a battery is defective given that it was classified as good = 1.92 %
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