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# Statistics 121 week 7 STAT 121

BYU

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This 5 page Class Notes was uploaded by Sydney Clark on Friday October 14, 2016. The Class Notes belongs to STAT 121 at Brigham Young University taught by Dr. Christopher Reese in Winter 2016. Since its upload, it has received 4 views.

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Date Created: 10/14/16

Stats notes *IMPORTANT* *EXAMPLE TEST QUESTION* The notation for mean and standard deviation is different for density curves (populations) than for histograms (samples). o DRAW THE PICTURE Observe the effect of changing mean and standard deviation on the Normal curve. Open StatCrunch Applet o “Distribution demos” Standard normal distribution o mean = 0, standard deviation = 1 o appropriate density curve for all normally distributed variables if converted to standard deviations from the mean e.g., prop. birth weights less than 3060 g = prop. of birth weights more than 1 standard deviation below the mean Proportion is NOT in percentage o In decimal Standarization o mathematical conversion of normally distributed variable to standard normal variable o if x is normally distributed with mean µ and standard deviation σ, then o Z-Score If z score is negative you are below the mean If z score is positive you are above the mean o birth weights of full-term babies are approximately normally distributed with µ = 3485 g and σ = 425 g o hence z = (weight−3485)/ 425 approximate standard normal variable Jennie scored 600 on the SAT Mathematics exam. Her friend Gerald took the American College Testing (ACT) test and scored 21 on the math part. ACT scores are normally distributed with mean 18 and standard deviation 6. SAT scores are also normally distributed with mean 500 and standard deviation 100. Assuming that both tests measure the same kind of ability, who has the higher score? o (a) Gerald o (b) Jennie o (c) Gerald’s and Jennie’s scores are the same o (d) Cannot be computed with information given normal density curve with µ = 0 and σ = 1 area under curve = proportion areas under curve for standard normal completely specified in the table of standard normal probabilities consists of z-scores or standardized scores with mean µ = 0 and σ = 1 to convert a normal random variable x to z: proportion of values greater than 1.5? o can’t determine using the “standard deviation rule” (i.e., the “68-95- 99.7 rule”) 2 Procedures: o standard normal table (the table of standard normal probabilities) • normal distribution function of statistical software Finding a proportion from a given X value o Assume that for a species of fruit flies, life length (X) in a lab follows a normal distribution with a mean of 53 days and a standard deviation of 15 days. When you’re asked questions like... “What proportion of flies live longer than 60 days?” “What proportion of flies live shorter than 30 days?” “What proportion of flies live between 20 and 40 days?” ... convert X to Z, then convert Z to a proportion. Finding an X value from a given proportion o Assume that for a species of fruit flies, life length (X) in a lab follows a normal distribution with a mean of 52 days and a standard deviation of 15 days. When you’re asked questions like... “80% of flies die before what age?” “64.5% of flies live beyond what age?” “What death ages (symmetric about the mean) includes two- thirds of all fruit flies?” ... convert proportion to Z, then convert Z to an X. Parameter: Numerical fact about the population (e.g., p=population proportion) Statistic: Corresponding numerical fact in the sample (e.g., pˆ=sample proportion) o 1. Parameters are typically unknown → (because it is usually impossible to know exactly what values a variable takes for every member of the population) 2. Statistics are computed from the sample → vary from sample to sample due to sampling variability o We want to understand how statistics behave relative to the parameter p = population proportion (parameter) pˆ = sample proportion (statistic) pˆ = number of individuals in category of interest/ number of individuals in sample The sampling distribution of a sample statistic refers to what the distribution of the statistic would look like if we chose a large number of samples from the same population The sampling distribution of a sample proportion is a theoretical probability distribution o It describes the distribution of: all sample proportions from all possible random samples of the same size taken from a population center: o mean= p o spread= o shape= approximately normal if n is large, but large depends on how close p is to 0.5. Guideline: np ≥ 10 and n(1 − p) ≥ 10

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