Statistics

Statistics: Inference

Descriptive measures, the normal distribution, and confidence intervals.

1. 1. Mean and spread

   easy

   A dataset is $\{4,8,6,10,12\}$. Compute the mean and the population standard deviation.

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   Mean: $\bar{x}=\frac{4+8+6+10+12}{5}=\frac{40}{5}=8$.

   Deviations squared: $(-4{)}^2,(0{)}^2,(-2{)}^2,(2{)}^2,(4{)}^2=16,0,4,4,16$, sum $=40$.

   Variance $=\frac{40}{5}=8$; standard deviation $=\sqrt{8}\approx 2.83$.

2. 2. Z-score and probability

   medium

   Test scores are normally distributed with mean $\mu =70$ and standard deviation $\sigma =10$. What fraction of students score above $85$? (Use $P(Z>1.5)\approx 0.0668$.)

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   $z=\frac{85-70}{10}=1.5$.

   $P(X>85)=P(Z>1.5)\approx 0.0668$, i.e. about 6.7% of students score above $85$.

3. 3. Confidence interval

   hard

   A sample of $n=100$ has mean $\bar{x}=50$. The population standard deviation is $\sigma =8$. Construct a $95\%$ confidence interval for the population mean.

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   Standard error $=\frac{\sigma }{\sqrt{n}}=\frac{8}{\sqrt{100}}=\frac{8}{10}=0.8$.

   Margin $=1.96\times 0.8=1.568$.

   Interval: $50\pm 1.568=(48.43,51.57)$. We are $95\%$ confident the true mean lies in this range.

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