Help Assignment 3 Part B All Students V3

Assignment 3 Part B FAQ HELP : Finding Z-scores: Example:To find Z 0.025 , (the z value such that 2.5% of the data lies above it) Commands: Distributions>Continuous Distributions>Normal Distribution>Normal Quantiles which returns so Z 0.025 = 1.959964 (that is, 2.5% of the data lies above the z value of 1.959964) A qth percentile is the z value such that the given percent q% of the data lies below it. So (1-q)% of the data lies above that z value. Example: What is the 43th percentile for a N(0,1) distribution? (this means 43% of the data lies below the 43th percentile, and 57% lies above it) You find Z 0.57 Finding normal areas under a curve with R and RCmdr DEMO QUESTION 10B IS HELPFUL (look at solution doc and command file doc side-by-side) Finding normal probabilities (areas under the curve) *percentage below: P(X < x) Commands: Distributions > Continuous Distributions>Normal Distribution>Normal Probabilities Variable Value(s): x value from problem Mean: mu from problem Standard Deviation: sigma from problem Lower Tail OK *percentage above: P(X > x) = 1 – P(X < x) and run R Commander commands to get P(X < x) *percentage between: P(a < X < b) = P(X < b) – P(X < a) and run R Commander commands twice to get P(X<b) and P(X<a) and then do the subtraction by hand. Finding Quantiles *keep in mind that a percentile is the X value such that the given percent of the data lies below it. *same as asking to find the x value such that some percent q% of the data lies below it. Example: Find the 20 th percentile for a X distribution that is N(1000, 40)

You want the X value where 0.20 of the data is below it (and 80% of data lies above it) You want unknown X = ? value where P(X < ?) = 0.20 (or you can think of it as z 0.080 ) Commands Distributions>Continuous Distributions>Normal Distribution>Normal Quantiles Probabilities: 0.20 Mean 1000 Standard deviation 40 Lower Tail, OK * find the value x such that some percent q% of the data lies above it Commands: Distributions > Continuous Distributions>Normal Distribution>Normal Quantiles Probabilities: given p (in decimal form) Mean: mu from problem Standard Deviation: sigma from problem Upper tail, OK Understanding a Quality Control Problem using rejection probability of a single unit from a Normal Distribution problem as the “success probability” in a Binomial Problem with n independent trials. Example : Consider a Normal distribution of grades with mean 65 & standard deviation 5. *A student with a grade below 55 receives an F. What percent of students receive an F? COMMANDS: Distributions>Continuous Distributions>Normal Distribution>Normal Probabilities Variable Values: 55 Mean 65 Standard deviation 5, OK Lower Tail, OK 2.27% of students fail *Randomly pick 5 students. What is the probability at least 1 student receives an F? This is a BINOMIAL problem with 5 trials, and probability of receiving an F is 2.27%. X is number of failures in 5 trials P(at least one F) = P(X > =1) = 1 – P(X <= 0). COMMANDS: Distributions>Discrete Distributions>Binomial Distribution>Binomial Tail Probabilities Variable Values: 0 Binomial Trials: 5 Probability of Success: 0.0227 Lower Tail, OK P(X>=1) = 1 – 0.8915373 = 0.1084627 ……………………………

QUESTIONS WHERE REPLICATED SAMPLES OF SIZE n ARE TAKEN AND THE SAMPLING DISTRIBUTION OF SAMPLE MEANS IS EXAMINED. Ignore parts of the demos where replications of samples of size n from a population are made to create a dataset that contains a column of sample means for each of those replications . Don’t try to do set.seed and sample and make datasets. Two Examples: Datasets have columns of sample means calculated from repeated samples of various sizes of n from a normal population (or an exponential population) are provided for you and put into one dataset. You use these columns to investigate the sampling distributions with those various sizes of n. Normal Parent Population Question: Use the dataset SampleMean_Normal.xlxs found via the datasets 3 link in the merged lab course. It has 3 columns:, x (column of observations from a normal population (for samples of size 1 from a normal population )), xbarn6 (column of sample means for samples of size 6 from a normal population), and xbarn36 (column of sample means for samples of size 36 from a normal population). For the x column, do the following commands. statistics > summaries > numerical summaries variable: x, OK graphs > histogram variable: x, OK Repeat these two commands for the xbar.n6 column. Repeat these two commands for the xbar.n36 column. Paste your R output. Demo 12 can be helpful. Exponential Parent Population Question: Use the dataset SampleMean_Exponential.xlxs via the datasets 3 link in the merged lab course. It has 3 columns: x (column of observations from an exponential distribution (for samples of size 1 from an exponential population) ), xbarn5 (column of sample means for samples of size 6 from an exponential distribution), and xbarn64 (column of sample means for samples of size 36 from an exponential distribution). For the x column, do the following commands. statistics > summaries > numerical summaries variable: x, OK graphs > histogram variable: x, OK Repeat these two commands for the xbar.n6 column. Repeat these two commands for the xbar.n36 column.

Paste your R output. The Exponential Parent Population Demo from the R manual can be helpful. Finding normal probabilities with a X ≈N(µ, σ √ n ) distribution DEMO 15 is relevant (look at solution doc and command file doc side-by-side) *Assume a random sample of size n is taken from a population with mean µ and standard deviation σ. * X is N(µ, σ √ n ) distribution. For given n, and #s, you will use R to find: * P(# 1 < X < # 2 ) = P( X < # 2 ) – P( X < # 1 ) for a N(µ, σ √ n ) distribution. * P( X < ¿¿ for a N(µ, σ √ n ) distribution. *P( X > ¿¿ for a N(µ, σ √ n ) distribution. If the population is normal, you can do this for any n. If the population is of unknown shape, you can only do this for large n (n above around 30 is good). COMMANDS to find P( X > ¿¿¿ Distributions>Continuous Distributions>Normal Distribution>Normal Probabilities Variable values(s): # Mean: mu Standard Deviation: σ √ n Upper Tail OK COMMANDS to find P( X < ¿¿¿ Distributions>Continuous Distributions>Normal Distribution>Normal Probabilities Variable values(s): # Mean: mu Standard Deviation: σ √ n Lower Tail OK