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7.5 – The Central Limit Theorem The x̅ distribution, given x is normal. x vs x̅ Theorem 7.1 for a Normal Probability Distribution Let x be a random variable with a normal distribution whose mean is µ and whose standard deviation is ơ. Let x̅ be the sample mean corresponding to random size n taken from the x distribution. The following true are true: 1. The x̅ distribution is a normal distribution. 2. The mean of the x̅ distribution is µ 3. The standard deviation of the x̅ distribution is ơ 𝑛 So… We can convert the x̅ distribution to the standard normal z distribution. µ𝑥̅ = µ ơ𝑥̅ = z = 𝑥̅ − µ ơ = ơ 𝑛 𝑥̅ − µ𝑥̅ ơ𝑥̅ = 𝑥̅ − µ ơ/ 𝑛 Vocabulary / Terminology 𝑺𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒆𝒓𝒓𝒐𝒓 𝒐𝒇 𝒕𝒉𝒆 𝒎𝒆𝒂𝒏 ∶ ơ𝑥̅ = ơ 𝑛 Central Limit Theorem 7.2: x can have any distribution, but as the sample size gets larger and larger, the distribution of x will approach a normal distribution. • Large sample: n ≥ 30 Guided Exercise #10 • As whole group, turn to page 290 – Look-over answers – Whole group clarification Find Probabilities Regarding x̅ 1. Must be normally distributed – If x distribution is normal, then x̅ is normal – If x distribution is not normal, but the sample size n ≥ 30, then x̅ is approximately normal 2. Convert x̅ to z using formula z = 𝑥̅ − µ𝑥̅ ơ𝑥̅ = 𝑥̅ − µ ơ/ 𝑛 3. Use the standard normal distribution (table 3) to find corresponding probabilities regarding x̅ Guided Exercise #11 • As whole group, turn to page 292 – Look-over answers – Whole group clarification Checkpoint You don’t look at anyways… Homework • Read Pages 286-293 – Take notes on what we have not covered • Do Problems – Page 293-296 (1-13) odds • Check odds in back of book • Read and preload 7.6 information – Notes/vocab