Abraham Baldwin Agricultural College Math 2000 Practice Test 3

Transcription

1 Abraham Baldwin Agricultural College Math 000 Practice Test 3 To get credit you must show your work. If your answer involves using a formula, make sure you write down the formula! Also write down any expression you enter into the calculator. Mark clearly whether you are using a z-distribution, a t-distribution, or a chisquared distribution. 1. In the standard normal distribution, find P( 0.3 z 1.65). Draw a picture and give the formula used to find the probability.. Find the value of z in the standard normal distribution such that the area to the right of z is Draw a picture and give the formula used to find the probability. 3. The mean time it takes students to complete a statistics exam is 40 minutes with a standard deviation of 5 minutes. Assuming the times are normally distributed, find the probability that a student completes the exam in less than 3 minutes. 4. Packages of ground beef in a supermarket have weights that are normally distributed with a mean weight of 16 ounces and a standard deviation of 0.08 ounces. What is the probability that a package will weigh between ounces and ounces? 5. The incredibly-concerned-about-profit-margins shoe company produces shoes whose life is normally distributed with a mean life of 10 months and a standard deviation of 1.3 months. They wish to offer a guarantee. How many months should the guarantee be if they do not wish to replace more than 5% of the shoes sold? 6. The mean weight of airline passengers is 154 pounds with a standard deviation of 15 pounds. Find the probability that the mean weight of 5 passengers on an airplane will exceed pounds. 7. A sample of 144 tomato plants in an agricultural experiment had an average yield of 5 pounds of tomatoes with a population standard deviation of 0.66 pounds. Find the 95% confidence interval for the mean yield of tomato plants. 8. A sample of 9 tomato plants in an agricultural experiment had an average yield of 5 pounds of tomatoes with a standard deviation of 0.66 pounds. Find the 90% confidence interval for the mean yield of tomato plants. 9. In a survey, 84 out of 140 people believed that statistics should be compulsory for people who want to understand and how the world works. Find the 99% confidence interval for the proportion of people who believe (quite correctly) that statistics should be compulsory for people who want to understand and how the world works.

2 10. The president of Dogdale College wishes to know the proportion of students who enjoy their math classes. An earlier survey showed that 78% of students enjoy their math classes. If the president wishes to know the percent of students who enjoy their math classes within 4 percentage points at the 90% confidence level, how many students should be surveyed? 11. It was found that the variance ( s ) in a sample of 17 measurements was 1.. Find the 98 percent confidence interval for the population variance. 1. One of every three Americans believes that the U.S. government should take primary responsibility for eliminating poverty in the U.S. If one hundred Americans were selected at random, find the probability that at most thirty of them will believe that the U.S. government should take primary responsibility for eliminating poverty in the U.S. SOLUTIONS: You will be asked to list what formula and points you used like in number 1,, and 3. You will also be asked what t-value, z-value, or chi-squared value you used %. 5. The guarantee should be 7 months (or less) p The sample size should be 9. 16(1.) 16(1.) 11., The probability that at most thirty of them will believe that the U.S. government should take primary responsibility for eliminating poverty in the U.S. is 7.43%

3 Abraham Baldwin Agricultural College Math 000 Test 3 Solutions 1. In the standard normal distribution, find P( 0.3 z 1.65) The problem is to find the shaded area in the diagram below The function that finds areas like the one shaded on the TI83 is normalcdf. Normalcdf(0.3, 1.65) = Find the value of z in the standard normal distribution such that the area to the right of z is The picture for this problem is as follows. Note that the area to the right of z is 0.45 and the area to the right of 0 is 0.5, so z will be just to the right of 0. The function that gives the value of z on the TI83 is invnorm. However, you must enter the area to the left of z on the calculator. The total area under the curve is 1, so the area to the left of z is = Invnorm(.55) = z = 0.13 z 3. The mean time it takes students to complete a statistics exam is 40 minutes with a standard deviation of 5 minutes. Assuming the times are normally distributed, find the probability that a student completes the exam in less than 3 minutes.

4 The problem is to find Px ( 3) in a normal distribution with mean 40 and standard deviation 5. x You convert to z-scores using z P( x 3) P( z ) P( z 1.6) 5 The problem is to find the shaded area in the picture. 1.6 You use Normalcdf on the TI83. To do so you must enter the left hand edge of the interval and the right hand edge. When you have no left hand edge use a large negative number, such as E99. = Normalcdf (E99, 1.6) = Packages of ground beef in a supermarket have weights that are normally distributed with a mean weight of 16 ounces and a standard deviation of 0.08 ounces. What is the probability that the packages will weigh between ounces and ounces? The problem is to find P(15.97 x16.04) in a normal distribution with mean 16 and standard deviation x You convert to z-scores using z P(15.97 x 16.4) P( z ) P(.375 z 0.5) The problem is to find the shaded area in the picture.

5 = Normalcdf (.38, 0.5) = You have just found that the probability that a package weighs between ounces and 16.4 ounces is The incredibly-concerned-about-profit-margins shoe company produces shoes whose life is normally distributed with a mean life of 10 months and a standard deviation of 1.3 months. They wish to offer a guarantee. How many months should the guarantee be if they do not wish to replace more than 5% of the shoes sold? The problem is to find a value, x, such that 5% of shoes have a life of x or less. That is, you want x such that P( x x ) 0.05 in a normal distribution with mean 10 and standard deviation 1.3. You first find the z-score for x and then convert to x using xz. The picture for z is: Shaded area = 0.05 z 0 z 0 = invnorm(0.05) = Then x z10 (1.3)( 1.65) The guarantee should be 7 months. 6. The mean weight of airline passengers is 154 pounds with a standard deviation of 15 pounds. Find the probability that the mean weight of 5 passengers on an airplane will exceed pounds.

6 z This problem involves the central limit theorem. Px ( 155.5) x Pz ( 1.5) = Normalcdf (1.5, E99) = n 5 The probability that the mean weight of 5 passengers on an airplane will exceed is A sample of 144 tomato plants in an agricultural experiment had an average yield of 5 pounds of tomatoes with a population standard deviation of 0.66 pounds. Find the 95% confidence interval for the mean yield of tomato plants. Since the sample standard deviation is given, the confidence interval is x z. For 95% confidence : You find z from the following n picture. Shaded area = = 0.05 z = invnorm(.05) = x z 5 (1.96)( ) n 144 The interval is 4.89 pounds to pounds. Answer: round to two decimal places to match given statistics. 8. A sample of 9 tomato plants in an agricultural experiment had an average yield of 5 pounds of tomatoes with a standard deviation of 0.66 pounds. Find the 90% confidence interval for the mean yield of tomato plants. Since the sample standard deviation is given, the confidence interval is

7 s x t. For 90% confidence : You find n with df = n 1 = 8 t from the t-table s.66 x t 5 (1.86)( ) n 9 The interval is pounds to pounds. Answer: round to two decimal places to match given statistics. 9. In a survey, 84 out of 140 people believed that statistics should be compulsory for people who want to understand and how the world works. Find the 99% confidence interval for the proportion of people who believe (quite correctly) that statistics should be compulsory for people who want to understand and how the world works. The formula is pˆ z pq ˆˆ. You can find n z using a picture similar to the one in Question 7 with : z = invnorm (0.995) =.576. You can also find 84 the last line of the table. pˆ 0.6 ; qˆ pˆ z in The interval is pq ˆˆ (0.6)(0.4) pˆ z n 140 The interval is to Answer:.493 p.707 round to three decimal places to match given statistics. 10. The president of Dogdale College wishes to know the proportion of students who enjoy their math classes. An earlier survey showed that 78% of students enjoy their math classes. If the president wishes to know the percent of students who enjoy their math classes within 4 percentage points at the 90% confidence level, how many students should be surveyed? The formula is z n pq ˆˆ E. The difficulty is trying to determine exactly what p is in the formula. Your options are to consider a worst case scenario and take p to be 0.5, or as, in this case, use an approximate value for p determined from an earlier sample. Four percentage points means E.04. For 90% confidence.10 and you find z as in Question 7. In this case z = 1.645

8 ˆp = 0.78; ˆq = 1 ˆp = 0.; z = 1.65 z 1.65 n pq ˆˆ (0.78)(0.) E 0.04 To be sure that n is large enough round up. The sample size should be It was found that the variance ( s ) in a sample of 17 measurements was 1.. Find the 98 percent confidence interval for the population variance. The formula for the confidence interval is ( n 1) s ( n 1) s right left 0.0 for 98% confidence. You use the row for df n 1 16 The value of right comes from the column under = 0.01 for 98% confidence. The value of left comes from the column under1 = 0.99 for 98% confidence. (The at the top of the table is misleading. It is not the same as the you get from the level of confidence.) right = and left = (1.) 16(1.) One of every three Americans believes that the U.S. government should take primary responsibility for eliminating poverty in the U.S. If one hundred Americans were selected at random, find the probability that at most thirty of them will believe that the U.S. government should take primary responsibility for eliminating poverty in the U.S. 1 p 3.33 q.67 n = z P( X 30) P( X 30.5) P( z.59) normalcdf(-e99,-.59) =.776 The probability that at most thirty of them will believe that the U.S. government should take primary responsibility for eliminating poverty in the U.S. is 7.76%