Section 05.1 and 05.2 shared lab

docx

School

Pennsylvania State University *

*We aren’t endorsed by this school

Course

200

Subject

Statistics

Date

Apr 3, 2024

Type

docx

Pages

Uploaded by ConstableMetal3677

LAB 5.1 & 5.2: APPROXIMATING WITH A DISTRIBUTION Statistics 200: Lab Activity for Sections 5.1 and 5.2 Learning objectives - Approximating with a distribution:  Recognize the shape of a normal distribution, and how the mean and standard deviation relate to the center and spread of a normal distribution  Recognize how the normal distribution can be used to approximate a bootstrap or randomization distribution for proportion(s)  Find exact areas in a normal distribution using technology  Compute a p-value or confidence interval for proportion(s) using a normal distribution  Find percentiles (sometimes called endpoints in the textbook) of a standard normal distribution and recognize how they can be used to find the multiplier for the confidence interval based on proportion(s) Confidence Interval Standardized test statistic Sample statistic ±(multiplier)×SE sample statistic − nullvalue SE Activity 1: Getting comfortable with the normal distribution Consider the following three hypothetical scenarios: A. The typical amount of sleep per day for very young babies has a bell-shaped distribution with a mean of 18 hours and a standard deviation equal to 1 hour. B. The amount of time spent playing video games per week for gamers in the US has a bell-shaped distribution with a mean of 7 hours and a standard deviation equal to 2 hours. C. The hours of work per week for a sample of CEOs has a bell-shaped distribution with a mean of 65 hours and a standard deviation of 5 hours. Consider the figure above. 1. Match the curves to the three scenarios found above: Curve 1- _____B____________, Curve 2 - _____A____________, Curve 3 - ____C____________ 2. Use StatKey to answer the questions below: What proportion of very young babies: 12/15/18 © - Pennsylvania State Univeresity

LAB 5.1 & 5.2: APPROXIMATING WITH A DISTRIBUTION a. sleep between 17 and 19 hours per day? 0.683 b. sleep between 16 and 20 hours per day? 0.954 c. sleep between 15 and 21 hours per day? 0.997 This is an illustration of the empirical rule, which states that if data are normal distributed approximately 68%, 95% or 99.7% falls within 1, 2, or 3 standard deviations of the mean respectively. Recall the below distributions:  The amount of time (hours) spent playing video games per week for gamers in the US: N(7,2)  CEO hours worked per week: N(65,5) 3. Use the empirical rule to quickly answer the questions below without using StatKey. What proportion of: a. US gamers spend between 3 and 11 hours playing video games per week? 0.954 b. US gamers spend between 1 and 13 hours playing video games per week? 0.997 c. CEOs work between 60 and 70 hours per week? 0.683 4. Draw a picture of a normal distribution with mean 40 and standard deviation 3, labeling the horizontal axis with the mean and standard deviation values that illustrate the Empirical Rule. Drawing normal distributions and p-values is a very useful skill in the coming weeks. Activity 2: Using the normal distribution to perform a hypothesis test for a single proportion We want to test H 0 : p = 0.5 vs H a : p > 0.5 using a sample proportion of 520 out of 1000, or 0.52. 1. Create a randomization distribution for this situation using at least 5,000 simulated samples. Where should this distribution be centered? At the null = 0.5 What is the standard error of the sample proportion? 0.016 2. Use your randomization distribution to find the p-value: 0.107 3. Model the randomization distribution with a normal distribution with mean 0.5 from the null hypothesis and standard deviation equal to the standard error of the sample proportion. Find the p-value using this normal distribution. 12/15/18 © - Pennsylvania State Univeresity

LAB 5.1 & 5.2: APPROXIMATING WITH A DISTRIBUTION 0.106 4. Find the standardized test statistic: (observed statistic – null value)/SE (0.52 – 0.5)/0.016 = 1.25 5. Find the p-value using the standardized test statistic and the standard normal distribution. 6. Compare the three p-values! What do you notice? The p-value of the normal distribution is smaller. Activity 3: Finding percentiles (sometimes called endpoints in the textbook) in a standard normal distribution: 1. Based on the pictures below, what is the correct z-multiplier (z*) for an 80% confidence interval? This is the correct z-multiplier for an 80% confidence interval 12/15/18 © - Pennsylvania State Univeresity

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

LAB 5.1 & 5.2: APPROXIMATING WITH A DISTRIBUTION 2. Use StatKey to find the z-multiplier for a 90% confidence interval. z* = 1.645 3. Use StatKey to find the z-multipliers for 95% and 99% confidence intervals. 95% = 1.960 99% = 2.576 4. Without using StatKey, only consider your answers to part 3 above. What is z* for a 96.5% confidence interval? a. z* = 2.108 b. z* = 2.765 c. z* = 1.812 Activity 4: Confidence interval for a difference in proportions We will use StatKey to calculate a confidence Interval for a difference in proportions using the StatKey dataset “Use Text Messages (by age)”, which compares text messaging use between teens and adults. 1. Create a bootstrap distribution of 4000 bootstrap differences of proportions using this sample. Use the "Two-tail" option to find the boundaries giving 95% in the middle of the bootstrap distribution. [0.120, 0.180] 2. What is the difference in sample proportions for this sample? 0.155 3. What is the standard error of this difference? 0.015 4. Switch to StatKey 's Normal distribution and edit the parameters so the mean is the sample statistic from question 2 and the standard error is the answer from question 3. Again, use the "Two-tail" option to find the boundaries giving 95% in the middle of this normal distribution. [0.126, 0.184] 12/15/18 © - Pennsylvania State Univeresity

LAB 5.1 & 5.2: APPROXIMATING WITH A DISTRIBUTION 5. Compare the boundaries from the normal distribution to the boundaries you found from the bootstrap distribution. Are the results similar? Yes the results are very similar, with the Normal distribution being a little bit higher. Activity 5: More practice making confidence intervals Gallup administered a poll to a random sample of 1,028 US adults between November 2 –8, 2017. Each participant was asked “if you were taking a new job and had your choice of a boss, would you prefer to work for a man or a woman?” 1. 55% of those sampled said they had no preference. The standard error of this sample proportion is 0.0155. Use the normal distribution to find a 98% confidence interval for the proportion of US adults who have no preference. a. What is z* ? 2.237 b. Find and interpret the 98% confidence interval. [0.514, 0.586] ?? 2. Using the same sample, the difference in sample proportions saying they had no preference was 0.24 when comparing men and women. The standard error of this difference is 0.030. Use the normal distribution to find a 90% confidence interval for the difference in proportions between the two groups: Note: N(0.24, 0.030) a. What is z* ? (Hint: already found this number in Activity 3) 1.645 b. Find the 90% confidence interval. [0.191, 0.289] ?? 12/15/18 © - Pennsylvania State Univeresity

Section 05.1 and 05.2 shared lab

Related Documents