A sample mean X is an unbiased estimate of the population mean

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week 11 Confidence Interval (tinterval) for a Mean (Inferential Ssidicn) Asample mean X is an unblased estimate of the population mean 7&' . confidence Intervals A confidence Interval uses the variability around a statistic to come up with an estimate for a rameter. Each confidence interval is associated with a confidence level. The confidence fevel €1 You how often the confidence interval will capture the parameter of interest. conditions for a t-interval for a Mean Sampling Distribytions of Sample Means Lettheré be a population with a true mean J' of some variable of interest. If taking samples of size L the sampling distribution of the sample mean _Y__ will satisfy the following pmpenies under the given conditions. Note: We do not know the population mean, so we will estimate it with the sample mean. Property (formula) Condition for Inference: and how to meet it i = M/ Random Sample o known: ,?:_ Independent trials: Each sample is an \)'VT independent event, drawn with replacement, OR o unknown: 0 the sample size, n, is less than or equal to Standal R 10% of the population size, N. &< 0.10 # utor 7 o known: Normal distribution (normaldist) Large sample or Normal population Critical Value is n 2= 30 (CLT) ¥ OR the original population distribution has an approximately symmetric distribution o unknown: Students' t distribution (tdist) Critical Value is
ff Example 1: Emily Is curious about the average lengih,of the female blue whales that rnlgf't':: n: the California coasl. She takes a Wo emale blue whales from this popula! % g Their average recorded length is ¥ = 75 m and their slandard deviation is s, = 11 m. The X ] distribution of lengths in the sample is roughly symmetric with no obvious oulfiers, Show that the 3 con ditions for inference, Random Sample, Independent Trials and Normal Approximation are salisfied. d l () fandonn samp C'l e . % = 1 '@ dgndtiht T talg ;o = poo 6005 ¢ o.(0 A i l @ VfilHMl AFPrDy'LY\ahOV\ YMQV" AlWWIhUV\ \( S\{W\.hnvr{'u( (/// sutlions | Students' t distribution &is vakhowly 3 + ditkib v i The Student's t-distributions (or simply t-distributions) are a family of distributions thak | represent the sampling distributions of the (-test statistics for the sample mean, 1, = i':! . n y when conditions are satisfied. These (-distributions have a similar shape (o the normal CUf"e ' but wider spreads and thicker tails, so more of the probability is contained in (he tails. This leads to higher critical values than those of the normal curve thus creating wider confidence intervals and higher p-values than normal distributions. The exact shape of a t-distribution is detcrmined by its degrees of freedom. The mare degrees of frecdom, the more similar the (-distribution is (o the normal distribution; in fact, you can interpret the normal distribution as a (-distribution with e degrees of freedom! Sample means from samples of size n will create a t-distribution with u | degrees of frecdom. theqrees ot fed Distribution oo " = - am o i Ghat | -l 0 fest statistic
T I R U 1 L M = —=y Hint: Use tdist( i df(percentile as a decimal) ) =h '\ - sample size of p = 22.V t?r:z(at:\e (:rilical value 1* for a 90% confidence level. (jf T A & : A7 |- 040 k" - H'"('W'qur(\f ((»ﬂO'rOO(}) 7 =6.10 1121 C)/ 2= 018 7 b Using a sample size of n = 14, find the critical value t* for a 95% confidence Ievel./_ ({1'0109 £¥: Higk (). inver el b (0949 ﬂ,()l%) /ﬂg '/.\\ y [F0-4% 2.l g +—= 0 ¥ - 006%9)~002% t= 3 4 ~- | 0 . ¢. Using degrees of freedom of 30, find the critical value t* for a 80% confidence level. = tdt ('36>~ e 46 (o 00't 0-1) L0 Example 3: Find the confidence level for each of the following critical values t: Hint: Use tdist(degrees of freedom) a. Using a sample size of n = 20, Susie made a one sample t-interval for a mean and used 0% 51 the critical value '=1.328. e \) = Adisk (1) /) R 10 a . a9 min - 1% may " [.518 AT ] cL= Bot -5y 0 b. Using a sample size of n = 34, Gerardo made a one sample t-interval for a mean-al used the critical value t*=1.474. (L it (1Y) i1 waag (L411) c. Using 7 degrees of freedom, Ciara made a one sample t-interval for a mean and used . the critical value t*=2.365 (O L—
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