STAT 1400 Lecture 3: 2.23 Stat Notes (Ch. 5)

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STAT 1400
Margaret Bryan

Stat 1400 2.23.2017 9:30 am In class examples: Workbook 5-4 Middle 95% Answer: 7.7 to 8.3 pH Workbook 5-5 a) .6736-.3264=.3472 Answer: 34.72% b) .9656-.0344=.9312 Answer: 93.12% c) Answer: .9312 d) .9967-.0033=.9934 Answer: .9934 The Central Limit Theorem Central limit theorem: When randomly sampling from any population with mean m and standard deviation s, when n is large enough, the sampling distribution of y̅ is approximately normal: N(m,s/√n). • The larger the sample size n, the better the approximation of normality. • This is very useful in inference: Many statistical tests require normality for the sampling distribution. The central limit theorem tells us that, if the sample size is large enough, we can safely make this assumption even if the raw data appear non-normal. How Large a Sample Size? It depends on the population distribution. More observations are required if the population distribution is far from normal. • A sample size of 25 or more is generally enough to obtain a normal sampling distribution from a skewed population, even with mild outliers in the sample. • A sample size of 40 or more will typically be good enough to overcome an extremely skewed population and mild outliers in the sample How do we know if the Population is Normal? • Sometimes we are told that a variable has an approximately normal distribution (e.g. large studies on human height or bone density). • Most of the time, we don’t know and we only have sample data. o We can summarize the data with a histogram and describe its shape. If the sample is random, the shape of the histogram should be similar to the shape of the population distribution. o If the histogram appears non-normal, the central limit theorem informs us whether the sampling distribution should look roughly normal or not. 1 Stat 1400 2.23.2017 9:30 am Population with strongly Sampling distribution of 𝑦 ത skewed distribution for n = 2 The larger the sample size (n) observations the more normal the distribution curve appears Sampling distribution of 𝑦 ത Sampling distribution ഥ for n = 10 of 𝑦 for n = 25 observations observations Examples:ram of big toe deformations 12 Angle of big toe deformations in 38 patients: • Symmetrical, one small outlier 10 • Population likely close to normal n 8 u 6 • Sampling distribution ~ normal r F 4 2 0 10 15 20 25 30 35 40 45 50 More HAV angle Histogram of # of fruit per day for 74 girls • Skewed, no outlier • Population likely skewed • Sampling distribution ~ normal given large sample size 2 Stat 1400 2.23.2017 9:30 am Atlantic acorn sizes (in cm ) 14 Sample of 28 acorns: 12 Describe the distribution of the sample. 10 Wha
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