PubH 6414 Worksheet 7b: Sampling Distribution of Sample Means 1 of 7

Transcription

1 PubH 6414 Worksheet 7b: Samplig Distributio of Sample Meas 1 of 7 Example 1: Calculatig Sample Sizes Assume that systolic blood pressure (SBP) for healthy adults is ormally distributed with = 1 mm Hg ad = mm Hg. a. What sample size is eeded so that 95% of sample meas are betwee 116 mmhg ad 124 mmhg? z-scores that correspod to the middle 95% of the area uder a stadard ormal distributio: ad We wat these z-scores to correspod to sample meas 116 ad 124 o the samplig distributio. Now use the z-score formula for either 116 or 124 to solve for (both will give the same aswer): * 1.96 * * roud up tosamplesize of 97 Due to roudig, more tha 95% of the area will be betwee 116 ad 124 at least 95% of the sample meas will be betwee 116 ad 124 if = 97. z-scores usig R ad Rcmdr: Rcmdr: Distributios > Cotiuous Distributios > Normal Distributio > Normal Quatiles Probabilities = 0.025, 0.975; select Lower Tail R script: qorm(c(0.025,0.975))

2 PubH 6414 Worksheet 7b: Samplig Distributio of Sample Meas 2 of 7 b. What sample size is eeded so that 90% of sample meas are betwee 116 mmhg ad 124 mmhg? First fid the z-scores that correspod to the middle 90% of the area uder a stadard ormal distributio. If 90% of the area is i the middle, there is 5% of the area i each of the two tails of the distributio. 90% of the area o the stadard ormal distributio is betwee ad Rcmdr: Distributios > Cotiuous Distributios > Normal Distributio > Normal Quatiles Probabilities = 0.05, 0.95; select Lower Tail R script: qorm(c(0.05,0.95)) Now use the z-score formula for either 116 or 124 to solve for (both will give the same aswer) * 1.645* * roud up tosamplesize of 68 Due to roudig, more tha 90% of the area will be betwee 116 ad 124 at least 90% of the sample meas will be betwee 116 ad 124 if = 68 c. What sample size is eeded so that SEM of the samplig distributio = 5? Use the formula for SEM to solve for SEM A sample size of 16 will result i a SEM = 5 whe the populatio stadard deviatio =.

3 PubH 6414 Worksheet 7b: Samplig Distributio of Sample Meas 3 of 7 d. What sample size is eeded so that SEM of the samplig distributio = 2? SEM A sample size of 100 will result i a SEM = 2 whe the populatio stadard deviatio = e. As sample size icreases the SEM of the samplig distributio decreases.

4 PubH 6414 Worksheet 7b: Samplig Distributio of Sample Meas 4 of 7 Example 2: Idetify the samplig distributio Idetify for each of the followig populatio examples whether the samplig distributio of the sample mea is distributed ormal with mea ad SEM t-distributio with df ad SEM = a. Based o log term data collectio mea ( Distributio of study hours per week is positively skewed. A sample of 60 olie studets has sample mea = 18.5 hours ad sample SD = 3.2 The samplig distributio of mea hours per week for samples of 60 studets is: Normal = b. Daily caloric itake for adolescet girls is ormally distributed with mea ( ) 20 kcal ad stadard deviatio ( ) 40 kcal. A sample of 36 adolescet girls has sample mea = 2210 kcal ad sample SD = 42 The samplig distributio of mea caloric itake for samples of 36 adolescet girls is: Normal 40/sqrt(36) = c. Childre betwee the ages of 2 ad 5 i the U.S. watch a average ( ) of 25 hours of televisio per week with ukow populatio stadard deviatio ( ). A sample of 49 childre age 2 to 5 has sample mea = 24 hours ad sample SD = 3.5. The samplig distributio of mea hours per week for samples of 49 childre age 2-5 is: a t-distributio with 48 df ad SEM = 3.5/sqrt(49) = 0.5 d. Mea ( ) BMI of maratho ruers is 22.4 but stadard deviatio is ukow. A sample of 100 maratho ruers has sample mea BMI = 22.7 ad sample stadard deviatio = 1.9. The samplig distributio of mea BMI for samples of 100 ruers is: a t-distributio with 99 df ad SEM = 1.9/sqrt(100) = 0.19 e. Mea ( ) BMI of maratho ruers is 22.4 but stadard deviatio is ukow. A sample of 49 maratho ruers has sample mea BMI = 22.8 ad sample stadard deviatio = 2.1The samplig distributio of mea BMI for samples of 49 ruers is: a t-distributio with 48 df ad SEM = 2.1/sqrt(49) = 0.3

5 PubH 6414 Worksheet 7b: Samplig Distributio of Sample Meas 5 of 7 Example 3: Calculatig Areas Uder the t-distributio Researchers are iterested i studyig systolic blood pressure (SBP) of urba police officers. It has bee determied that mea ( ) SBP i this populatio = 1 mmhg but stadard deviatio ( ) is ukow. For each of the followig idetify the samplig distributio a. A sample of 49 urba police officers has sample mea SBP = 117 with sample stadard deviatio = 18. i. Idetify the samplig distributio The samplig distributio of sample meas for samples of 49 officers is distributed t 48 with SEM = 18/7 = 2.57 ii. Calculate P ( x 117). Iterpret. t_117 = (117-1)/(18/7) = df = 48 Area below t_117 = Iterpretatio: The probability that a sample of 49 has a mea SBP less tha or equal to 117 whe the populatio mea is 1 = Rcmdr: Distributios > Cotiuous Distributios > t distributio > t Variable Value = t_117; df=48; select Lower Tail R Script: pt(t_117, df=48) b. A differet sample of 49 urba police officers has sample mea SBP = 126 with sample stadard deviatio =. i. Idetify the samplig distributio. The samplig distributio of sample meas for samples of 49 officers is distributed t 48 with SEM = /7 = 2.86 ii. Calculate P ( x 126). Iterpret. t_126 = (126-1)/(/7) = 2.1 df = 48 Area above t_126 = Iterpretatio: The probability that a sample of 49 has a mea SBP greater tha or equal to 126 whe the populatio mea is 1 is 0.02 Rcmdr: Distributios > Cotiuous Distributios > t distributio > t Variable Value = t_126; df=48; select Upper Tail R Script: pt(t_126, df=48,lower.tail=false) 1-pt(t_126,df=48)

6 PubH 6414 Worksheet 7b: Samplig Distributio of Sample Meas 6 of 7 c. A sample of 64 urba police officers has sample mea SBP = 116 ad sample stadard deviatio = 16. i. Idetify the samplig distributio. The samplig distributio of sample meas for samples of 64 officers is distributed t 63 with SEM = 16/8 = 2. ii. Calculate P ( x 116). Iterpret. t_116 = (116-1)/(2) = -2 df = 63 Area below t_116 = Iterpretatio: The probability that a sample of 64 has a mea SBP less tha or equal to 116 whe the populatio mea is 1 is Rcmdr: Distributios > Cotiuous Distributios > t distributio > t Variable Value = -2; df=63; select Lower Tail R Script: pt(-2, df=63) d. A differet sample of 64 urba police officers has sample mea SBP = 118 ad sample stadard deviatio = 16. i. Idetify the samplig distributio. The samplig distributio of sample meas for samples of 64 officers is distributed t 63 with SEM = 16/8 = 2. ii. Calculate P ( x 118). Iterpret. t_118 = (118-1)/2 = -1 df = 63 Area below 118 = Iterpretatio: The probability that a sample of 64 has a mea SBP less tha or equal to 118 whe the populatio mea is 1 is Rcmdr: Distributios > Cotiuous Distributios > t distributio > t Variable Value = -1; df=63; select Lower Tail R Script: pt(-1, df=63)

7 PubH 6414 Worksheet 7b: Samplig Distributio of Sample Meas 7 of 7 Take some time to compare the results for 3c ad 3d. Both of these examples have a sample of 64 with the same SEM ad a sample mea that is less tha the populatio mea. I 3c the sample mea SBP is 116 mmhg ad i 3d the sample mea SBP is 118 mmhg. Which is further from the true populatio mea of 1 mmhg? Which has a larger t-coefficiet (i absolute value)? Does it make sese that the probability of havig a mea SBP less tha or equal to 116 mmhg is smaller tha the probability of havig a mea SBP less tha or equal to 118 mmhg if the true populatio mea = 1 mmhg (give that sample size ad SEM are the same for the two samples)? Yes All other coditios (sample size ad SEM) beig the same: Larger differeces betwee the sample mea ad the populatio mea result i larger (absolute value) t-coefficiets Larger t-coefficiets are further from 0, the ceter of the t-distributio Therefore, the tail area beyod larger t-coefficiets is smaller o The tail area beyod (to the left of) the t-coefficiet is equal to the probability of observig the correspodig sample mea or a smaller sample mea whe the populatio mea is kow. So, there is smaller probability of observig a sample mea (or smaller) whe the sample mea is further from the populatio mea. The same argumets ca be applied to situatios where sample meas are larger tha the populatio mea: a larger differece betwee a observed sample mea ad the populatio mea results i a larger t-coefficiet ad a smaller probability of observig that sample mea or a larger sample mea (give costat sample size ad SEM). I 3c, the probability of observig a sample mea less tha or equal to 116 mmhg i a sample of 64 whe the true populatio mea is 1 mmhg is very small: This small probability idicates that it is ulikely to observe a sample mea of 116 mmhg or smaller if the true populatio mea is 1 mmhg. What are some possible explaatios for this ulikely sample mea? Perhaps this sample is ot represetative of the populatio. If the sample was t radomly selected, it may be represetig a subset of the populatio of urba police officers that has lower mea SBP o Maybe the samplig procedure was biased towards youger urba police officers with lower SBP, o average, resultig i this lower sample mea SBP. Perhaps the equipmet used to measure SBP i this sample was calibrated icorrectly ad the observed SBP is lower tha the actual SBP for each officer i the sample resultig i a smaller sample mea SBP. Perhaps the kow mea SBP i the populatio of urba police officers is ot actually = 1 mmhg as has bee stated. o Lesso 8 Part 2 covers hypothesis testig which is a procedure used to test whether sample data provide sigificat evidece agaist some statemet about the populatio from which the data were sampled.