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We can do this but need a modication to make it work

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Unformatted text preview: glu (UMich) Interval Estimation 7 / 16 Interval Estimation of Population Mean Case 2: Normally distributed population, σ unknown We keep talking about unknown population parameters: µ is unknown, we are building an interval estimate for it. How realistic is that σ is known. Not really: we almost never know what σ is. So what do we do? We can use s , sample standard deviation, instead. We can do this, but need a modiﬁcation to make it work. It turns out the sampling distribution of x is not normal if you use s instead of σ . ¯ Sampling Distribution of x with unknown σ ¯ If x is the sample mean of a sample drawn from a population with N (µ, σ 2 ) distribution, ¯ then t= x −µ ¯ √ s/ n has a Student’s t distribution with n-1 degrees of freedom Utku Suleymanoglu (UMich) Interval Estimation 8 / 16 Interval Estimation of Population Mean Student’s t-distribution is another continuous probability distribution. It looks a lot like the standard normal distribution with symmetricity and mean at zero. It has a single parameter, called its degrees of freedom. As df increases, it converges to N (0, 1). We will rely on another table to calculate probabilities for it. Utku Suleymanoglu (UMich) Interval Estimation 9 / 16 Interval Estimation of Population Mean Utku Suleymanoglu (UMich) Interval Estimation 10 / 16 Interval Estimation of Population Mean CI for µ, Case 2: Normal Population, σ unknown If population has a normal distribution and σ is unknown, but sample standard deviation,s , is known, a 100(1 − α)% conﬁdence interval for µ can be constructed via: √ x ±...
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