1、The Population Mean and Standard Deviation,1,X,Computing the Mean and the Standard Deviation in Excel, = AVERAGE(range) = STDEV(range),2,Exercise,Compute the mean, standard deviation, and variance for the following data: 1 2 3 3 4 8 10 Check Figures Mean = 4.428571 Standard deviation = 3.309438 Vari
2、ance = 10.95238,3,The Normal Distribution,4,X,P(- to X),Solving for P(- to X) in Excel,P(- to X) = NORMDIST(X, mean, stdev, cumulative) X = value for which we want P(- to X) Mean = Stdev = Cumulative = True (It just is),5,Exercise in Solving for P(- to X),What portion of the adult population is unde
3、r 6 feet tall if the mean for the population is 5 feet and the standard deviation is 1 foot? Check figure = 0.841345,6,P(X to ),7,X,P(X to ),P(X to ),P(X to ) = 1 P(- to X),8,X,P(X to ),P(- to X),P=1.0,Exercise,What portion of the adult population is OVER 6 feet tall if the mean for the population i
4、s 5 feet and the standard deviation is 1 foot? Check figure = 0.158655,9,P(X1 to X2),10,X1,P(X1 X X2),X2,P(X1 to X2) in Excel,P(X1 to X2) = P(- to X2) - P(- to X1) P(X1 to X2)=NORMDIST(X2)NORMDIST(X1),11,Exercise in P(X1 to X2) in Excel,What portion of the adult population is between 6 and 7 feet ta
5、ll if the mean for the population is 5 feet and the standard deviation is 1 foot? Check figure = 0.135905,12,Computing X,13,X,P(- to X),Computing X in Excel,X = NORMINV(probability, mean, stdev) Probability is P(- to X),14,Exercise in Computing X in Excel,An adult population has a mean of 5 feet and
6、 a standard deviation is 1 foot. Seventy-five percent of the people are shorter than what height? Check figure = 5.67449,15,Z Distribution,A transformation of normal distributions into a standard form with a mean of 0 and a standard deviation of 1. It is sometimes useful.,16,Z,0.12,0,X,8.6,8, = 8 =
7、10, = 0 = 1,P(X 8.6),P(Z 0.12),Computing P(- to Z) in Excel,Z = (X-)/ P(- to Z) = NORMDIST(Z, mean, stdev, cumulative) Mean = 0 Stdev = 1 Z = (X-)/ Cumulative = True (It just is),17,Exercise in Computing P(- to Z) in Excel,An adult population has a mean of 5 feet and a standard deviation is 1 foot.
8、Compute the Z value for 4.5 feet all. What portion of all people are under 4.5 feet tall Z check figure = -.5 (the minus is important) P check figure = 0.308537539,18,Z Distribution,A transformation of normal distributions into a standard form with a mean of 0 and a standard deviation of 1. It is so
9、metimes useful.,19,Z,0.12,0,X,8.6,8, = 8 = 10, = 0 = 1,P(X 8.6),P(Z 0.12),Computing Z in Excel,Z for a certain value of P(- to Z) =NORMINV(probilility, mean, stdev) Probability = P(- to Z) Mean = 0 Stdev = 1 Change the Z value to an X value if necessary Z = (X-)/, so X = + Z ,20,Exercise in Computin
10、g Z in Excel,An adult population has a mean of 5 feet and a standard deviation is 1 foot. 25% of the population is greater than what height? Check figure for Z = 0.67449 Check figure for X = 0.308537539,21,Sampling Distribution of the Mean,22,Normal Population Distribution,Normal Sampling Distributi
11、on (has the same mean), is the Population Standard Deviation,Xbar is the Sample Standard Deviation. Xbar = /n Xbar ,Sampling Distribution of the Mean,For the sampling distribution of the mean. The mean of the sampling distribution is Xbar The standard deviation of the sampling distribution of the me
12、an, Xbar, is /n This only works if is known, of course.,23,Exercise in Using Excel in the Sampling Distribution of the Mean,The sample mean is 7. The population standard distribution is 3. The sample size is 100 Compute the probability that the true mean is less than 5. Compute the probability that
13、the true mean is 3 to 5,24,Confidence Interval if is Known,Using X,25,Point Estimate for Xbar,Lower Confidence Limit Xmin,Upper Confidence Limit Xmax,X units:,Confidence Interval,95% confidence level Xmin is for P(- to Xmin) = 0.025 Xmax is for P(- to Xmax) = 0.975 X = NORMINV(probability, mean, std
14、dev) Here, stdev is Xbar = /n,26,Exercise,For a sample of 25, the sample mean is 100. The population standard deviation is 50. What is the standard deviation of the sampling distribution? Check figure: 10 What are the limits of the 95% confidence level? Check figure for minimum: 80.40036015 Check fi
15、gure for maximum: 119.5996,27,Confidence Interval if is Known,Done Using Z,28,Z/2 = -1.96,Z/2 = 1.96,Z units:,0,Confidence Intervals with Z in Excel,Xmin = Xbar Z/2 * /n Why? Because multiplying a Z value by /n gives the X value associated with the Z value Xmax = Xbar + Z/2 * /n Common Z/2 value: 95
16、% confidence level = 1.96,29,Exercise in Confidence Intervals with Z in Excel,The sampling mean Xbar is 100. The population standard deviation, , is 50. The sample size is 25. What are Xmin and Xmax for the 95% confidence level? Check figure: Z/2 = 1.96 Xmin = 80.4 (same as before) Xmax = 119.6 (sam
17、e as before),30,Confidence Intervals, Unknown,Use the sample standard deviation S instead of Xbar. No need to divide S by the square root of n Because S is not based on the population Use the t distribution instead of the normal distribution.,31,Computing the t values,Z = TINV(probability, df) proba
18、bility is P(- to X) df = degrees of freedom = n-1 for the sampling distribution of the mean. Xmin = Xbar Z(.025,n-1)*S Xmax = Xbar + Z(.975,n-1)*S,32,Exercise,For a sample of 25, the sample mean is 100. The sample standard deviation is 5. What is Z for the 95% confidence interval? Check figure 2.390
19、949 What is the lower X limit? Check figure 88.04525 (With known, was 80.40036015) What is the upper X limit? Check figure 111.9547 (With known, was 119.5996),33,t test for two samples,What is the probability that two samples have the same mean?,34,The t Test Analysis,Go to the Data tab Click on dat
20、a analysis Select t-Test for Two-Sample(s) with Equal Variance,35,With Our Data and .05 Confidence Level,36,t stat = 0.08t critical for two-tail (H1 = not equal) = 2.18.T stat t Critical, so do not reject the null hypothesis of equal means.Also, is 0.94, which is far larger than .05,t Test: Two-Samp
21、le, Equal Variance,If the variances of the two samples are believed to be the same, use this option. It is the strongest t testmost likely to reject the null hypothesis of equality if the means really are different.,37,t Test: Two-Sample, Unequal Variance,Does not require equal variances Use if you
22、know they are unequal Use is you do not feel that you should assume equality You lose some discriminatory power Slightly less likely to reject the null hypothesis of equality if it is true,38,t Test: Two-Sample, Paired,In the sampling, the each value in one distribution is paired with a value in the
23、 other distribution on some basis. For example, equal ability on some skill.,39,z Test for Two Sample Means,Population standard deviation is unknown. Must compute the sample variances.,40,z test,Data tab Data analysisz test sample for two means,41,Z value is greater than z Critical for two tails (not equal), so reject the null hypothesis of the means being equal. Also, = 2.31109E-08 .05, so reject.,Exercise,Repeat the analysis above.,42,
