Question

Problem 1: Confidence Interval for Percentage of B’s.

The data set “STAT 250 Final Exam Scores” contains a random sample of 269 STAT 250 students’ final exam scores (maximum of 80) collected over the past two years. Answer the following questions using this data set.

a) What proportion of students in our sample earned B’s on the final exam? A letter grade of B is obtained with a score of between 64 and 71 inclusive. Hint: You can do this many ways, but in StatCrunch, go to Data à Row Selection à Interactive Tools. In the slider selectors box, click the variable “Scores” into the variable box. Then click compute. Use the slider to obtain the count by looking at the “# rows selected” presented in the first line of the box. Show your work (i.e. describe the method you used to obtain the number of B’s) and express this value as a proportion rounded to four decimal places.

b) Using the sample proportion obtained in (a), construct a 90% confidence interval to estimate the population proportion of students who earned a B on the final exam. Please do this “by hand” using the formula and showing your work (please type your work, no images accepted here). Assume all Central Limit Theorem conditions hold. Round your confidence limits to four decimal places.

c) Verify your result from part (b) using Stat à Proportions Stats à One Sample à With Summary. Inside the box, select confidence interval and click Compute! Copy and paste your StatCrunch result in your document.

d) Interpret the StatCrunch confidence interval in part (c) in one sentence using the context of the question.

e) Did this confidence interval capture the true population proportion p = 0.21 given in Problem 4 of Data Analysis 2. Answer this question in one sentence.

f) Use the Confidence Interval applet (for a Proportion) in StatCrunch to simulate constructing one thousand 90% confidence intervals using p = 0.21 and n = 269. Once the window is open, click reset and select (or click) 1000 intervals. Copy and paste your image into your document.

g) Compare the “Prop. contained” value from part (f) to the confidence level associated with the simulation.

h) Write a long-run interpretation for your confidence interval method in one sentence.

Data:

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Answer 1

As per HomeworkLib policy only 4 question is answered.

Since the sample as

80, 80, 79, 78, 78, 77, 76, 76, 76, 76, 76, 75, 75, 75, 74, 74, 74, 73, 73, 72, 72, 72, 72, 72, 72, 71, 71, 71, 71, 71, 71, 70, 70, 70, 70, 70, 69, 69, 69, 68, 68, 68, 68, 68, 67, 67, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 64, 64, 64, 64, 64, 64, 64, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 62, 62, 62, 62, 62, 62, 62, 62, 62, 61, 61, 61, 61, 61, 61, 61, 60, 60, 60, 60, 60, 60, 60, 59, 59, 59, 59, 59, 59, 58, 58, 58, 58, 58, 58, 58, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 55, 55, 55, 55, 55, 55, 55, 55, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 53, 53, 53, 53, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 51, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 49, 49, 49, 49, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 46, 46, 46, 45, 45, 45, 45, 44, 44, 44, 44, 44, 43, 43, 43, 42, 42, 42, 42, 42, 42, 42, 42, 42, 41, 40, 40, 40, 39, 39, 39, 38, 37, 37, 37, 36, 36, 36, 36, 35, 35, 35, 34, 34, 34, 33, 32, 31, 30, 29

Mean=56.2565

Sample standard deviation=11.2288

a) Since 51 scored B hence proportion scored B as

51/269=0.1896

b)

Standard error of the mean = SEM = √x(N-x)/N³ = 0.024

α = (1-CL)/2 = 0.050

Standard normal deviate for α = Z_α = 1.645

Lower bound = P - (Z_α*SEM) = 0.1501

Upper bound = P + (Z_α*SEM) = 0.2291

c) Could not understand what actually question is wether it is hypothesis test or something else. Please write the correct question in comment box .

d) We can be 90 % confident that the proportion of B grade for population lies between (o.1501 to 0.2291)