STATISTICS 448N-66079Statistics_Binomial Distribution./ complete solution

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Use the binomial distribution to compute probability Question Jamie is practicing free throws before her next basketball game. The probability that she makes each shot is 0.6. If she takes 10 shots,... what is the probability that she makes exactly 7 of them? Round your answer to three decimal places. Select the correct answer below: A weighted coin has a 0.55 probability of landing on heads. If you toss the coin 14 times, what is the probability of getting heads exactly 9 times? (Round your answer to 3 decimal places if necessary.) 0.170 Using the Calculator to Determine the Probability of a Binomial Distribution If you have a graphing calculator available, like the TI-84, you can easily compute the binomial probability, B(x;n,p), where B denotes the distribution as binomial. Knowing the parameters of the binomial distribution is necessary, so remember: • n = the number of trials • x = the number of successes in the whole experiment • p = the probability of a success Let's use the same question as above as an example. Suppose a dice is rolled 5 times. A "success" is determined as rolling a 4. What is the probability of getting exactly two successes? The values of n, p, and x are: • n = 5 trials • x = 2 successes • p = 16≈0.167 Once the [DISTR] menu is opened, and the binompdf( function is selected, enter the values 5,(16),2 and close the parentheses as shown below: binompdf(5,(16),2) Then press ENTER. The calculator will show the probability of getting 4 exactly two times when the dice is rolled 5 times, which is approximately 0.16 A computer graphics card manufacturer is testing an improvement to its production process. If a sample of 100 graphics cards manufactured using the new process has a less than 10% chance of having 3 or more defective graphics cards, then the manufacturer will switch to the new process. Otherwise, the manufacturer will stay with its existing process. If the probability of a defective graphics card using the new process is 0.9%, will the manufacturer switch to the new production process? Yes, because the probability of having 3 or more defective graphics cards is greater than 0.10. Yes, because the probability of having 3 or more defective graphics cards is less than 0.10.xx No, because the probability of having 3 or more defective graphics cards is less than 0.10. No, because the probability of having 3 or more defective graphics cards is greater than 0.10. Note that this is a cumulative binomial probability. In this case, we want to find the probability of 3 or more successes, inclusive, where a success is one of the graphics cards being defective. The probability of having 2 or fewer defective graphics cards is the complement of the probability of having 3 or more defective graphics cards. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 2, 100, and 0.009, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.937964, which is 0.938 rounded to three decimal places. To find the probability of having 3 or more defective graphics cards, subtract this probability from 1. The probability of having 3 or more defective graphics cards is 1−0.938=0.062, which is less than 0.10. So, the manufacturer will switch to the new process. Question A roulette wheel has 38 slots, numbered 1 to 36, with two additional green slots labeled 0 and 00. Jim, a dealer at a roulette table, tests the roulette wheel by spinning a ball around the wheel repeatedly and seeing where the ball lands. The ball has an equally likely chance of landing in each slot. If Jim spins the ball around the wheel 25 times, what is the probability that the ball lands in a green slot at most twice? Use Excel to find the probability. • Round your answer to three decimal places. Answer Explanation Correct answers: Note that this is a binomial probability. In this case, we want to find the probability of 0 to 2 successes, inclusive, where a success is the ball landing in 0 or 00. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 2, 25, and 2/38, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.857891, which is 0.858 rounded to three decimal places. Create and interpret a Binomial Distribution with Technology - Excel Question A certain cold remedy has an 88% rate of success of reducing symptoms within 24 hours. Find the probability that in a random sample of 45 people who took the remedy, 40 of them had a reduction of symptoms within a day. • Round your answer to three decimal places. Answer Explanation Correct answers: Note that this is a binomial probability. In this case, we want to find the probability of 40 successes, where a success is a reduction of symptoms. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 40, 45, and 0.88, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.18289, which is 0.183 rounded to three decimal places. Question A state lottery sells instant-lottery scratch tickets. 12% of the tickets have prizes. Neil goes to the store and buys 10 tickets. What is the probability that exactly three of Neil's tickets will have prizes? Use Excel to find the probability. • Round your answer to three decimal places. $$0.085 Answer Explanation Correct answers: • $0.085$0.085 Note that this is a binomial probability. In this case, we want to find the probability of 3 successes, where a success is a ticket having a prize. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, and the probability of a success. In this case, enter 3, 10, and 0.12, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.084743, which is 0.085 rounded to three decimal places. Question In a large city’s recent mayoral election, 126,519 out of 283,143 registered voters actually turned out to vote. If 20 registered voters are randomly selected, find the probability that exactly 8 of them voted in the mayoral election. Use Excel to find the probability. • Round your answer to three decimal places. $$0.164 Answer Explanation Correct answers: • $0.164$0.164 Note that this is a binomial probability. In this case, we want to find the probability of exactly 8 successes, where a success is a registered voter turning out to vote. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 8, 20, and 126519/283143, in that order. In the entry for a cumulative probability, enter 0 since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.164322, which is 0.164 rounded to three decimal places. A statistics professor recently graded final exams for students in her introductory statistics course. In a review of her grading, she found the mean score out of 100 points was a x¯=77, with a margin of error of 10. Construct a confidence interval for the mean score (out of 100 points) on the final exam. QUESTION 3 The pages per book in a library are normally distributed with an unknown population mean. A random sample of books is taken and results in a 95% confidence interval of (237,293) pages. What is the correct interpretation of the 95% confidence interval? Select the correct answer below: _We estimate with 95% confidence that the sample mean is between 237 and 293 pages. _We estimate that 95% of the time a book is selected, there will be between 237 and 293 pages. _We estimate with 95% confidence that the true population mean is between 237 and 293 pages. QUESTION 4 The population standard deviation for the heights of dogs, in inches, in a city is 3.7 inches. If we want to be 95% confident that the sample mean is within 2 inches of the true population mean, what is the minimum sample size that can be taken? z0.101.282 z0.051.645 z0.0251.960 z0.012.326 z0.0052.576 Use the table above for the z-score, and be sure to round up to the nearest integer. [Show More]

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