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The Population Mean and Standard Deviation σ μ X 1 Computing the Mean and the Standard Deviation in Excel • μ = AVERAGE(range) • δ = 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 – Variance = 10.95238 3 The Normal Distribution P(-∞ to X) μ X 4 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 = µ – Stdev = δ – Cumulative = True (It just is) 5 Exercise in Solving for P(-∞ to X) • What portion of the adult population is under 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 ∞) P(X to ∞) μ X 7 P(X to ∞) • P(X to ∞) = 1 – P(-∞ to X) P(-∞ to X) P(X to ∞) μ P=1.0 X 8 Exercise • What portion of the adult population is OVER 6 feet tall if the mean for the population is 5 feet and the standard deviation is 1 foot? – Check figure = 0.158655 9 P(X1 to X2) P(X1 < X < X2) X1 X2 10 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 tall if the mean for the population is 5 feet and the standard deviation is 1 foot? – Check figure = 0.135905 12 Computing X P(-∞ to X) μ X 13 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 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. μ=8 σ = 10 8 8.6 P(X < 8.6) μ=0 σ=1 X 0 0.12 Z P(Z < 0.12) 16 Computing P(-∞ to Z) in Excel • Z = (X-μ)/δ • P(-∞ to Z) = NORMDIST(Z, mean, stdev, cumulative) (X – Mean = 0 – Stdev = 1 Z ) – Z = (X-μ)/δ – 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. 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 sometimes useful. μ=8 σ = 10 8 8.6 P(X < 8.6) μ=0 σ=1 X 0 0.12 Z P(Z < 0.12) 19 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-μ)/δ, so –X=µ+Zδ X μ Zσ 20 Exercise in Computing 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 Normal Population Distribution μ Normal Sampling Distribution (has the same mean) x δ is the Population Standard Deviation δXbar is the Sample Standard Deviation. μx x δXbar = δ/√n δXbar << δ 22 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 mean, δXbar, is δ/√n • This only works if δ 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 the true mean is 3 to 5 24 Confidence Interval if δ is Known • Using X 1 α 0.95 so α 0.05 α 0.025 2 X units: α 0.025 2 Lower Confidence Limit Xmin Point Estimate for Xbar Upper Confidence Limit Xmax 25 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, stddev) – Here, stdev is δXbar = δ/√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 figure for maximum: 119.5996 27 Confidence Interval if δ is Known • Done Using Z 1 α 0.95 so α 0.05 α 0.025 2 Z units: α 0.025 2 Zα/2 = -1.96 0 Zα/2 = 1.96 28 Confidence Intervals with Z in Excel • Xmin = Xbar – Zα/2 * δ/√n – Why? – Because multiplying a Z value by δ/√n gives the X value associated with the Z value • Xmax = Xbar + Zα/2 * δ/√n • Common Zα/2 value: – 95% confidence level = 1.96 29 Exercise in Confidence Intervals with Z in Excel • The sampling mean Xbar is 100. The population standard deviation, δ, is 50. The sample size is 25. What are Xmin and Xmax for the 95% confidence level? – Check figure: Zα/2 = 1.96 – Xmin = 80.4 (same as before) – Xmax = 119.6 (same as before) 30 Confidence Intervals, δ Unknown • Use the sample standard deviation S instead of δXbar. – No need to divide S by the square root of n – Because S is not based on the population δ • Use the t distribution instead of the normal distribution. 31 Computing the t values • Z = TINV(probability, df) – probability 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 2.390949 • What is the lower X limit? – Check figure 88.04525 (With δ known, was 80.40036015) • What is the upper X limit? – Check figure 111.9547 (With δ known, was 119.5996) 33 t test for two samples • What is the probability that two samples have the same mean? Sample Mean Sample A 1 3 Sample B 1 2 5 5 7 5 4 8 9 10 5.714286 9 10 5.571429 34 The t Test Analysis • Go to the Data tab • Click on data analysis • Select t-Test for TwoSample(s) with Equal Variance 35 With Our Data and .05 Confidence Level t stat = 0.08 t critical for twotail (H1 = not equal) = 2.18. T stat < t Critical, so do not reject the null hypothesis of equal means. Also, α is 0.94, which is far larger than .05 36 t Test: Two-Sample, Equal Variance • If the variances of the two samples are believed to be the same, use this option. • It is the strongest t test—most 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 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 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 analysis • z test sample for two means Z value is greater than z Critical for two tails (not equal), so reject the null hypothesis of the means being equal. Also, α = 2.31109E-08 < .05, so reject. 41 Exercise • Repeat the analysis above. 42