Six Sigma Practice Test 7 ( Indiana Council /Analyze Stage)
A limousine service wants to find the most consistent route between the city center and the airport to ease scheduling conflicts. Six timed runs were made along route A and the standard deviation was 10.5 minutes. Along route B, 8 test runs were made and the standard deviation dropped to 5.2 minutes. Determine whether there is sufficient evidence to determine that route B is more consistent with 95% confidence. What is the calculated and critical test values and what was the result?
A limousine service wants to find the most consistent route between the city center and the airport to ease scheduling conflicts. Six timed runs were made along route A and the standard deviation was 10.5 minutes. Along route B, 8 test runs were made and the standard deviation dropped to 5.2 minutes. Determine whether there is sufficient evidence to determine that route B is more consistent with 95% confidence. What is the calculated and critical test values and what was the result? Answer= Critical = 3.58 Calcuated = 4.08 Fail to reject the null Ho--route B is more consistent This is statistical inference problem using F-test. One is interested only in an improvement in variation (standard deviation) so a one-tail test, with all of the alpha risk in the right trail is requaired. Route A Route B No. of Tests 6 8 Standard 10.5 min 5.2 min deviation Fcalc is outside of F crit, so the null is rejected and we can conclude that Route B is more consistent than Route A. DF(A) = V(A) -5; DF(B) = V(B) -7; alpha = 0.05 F cirt= 3.97 (table) F calc= ( SD --a) ^2 / (Sd--b ^2) = 10.5 ^2 / 5.2 ^2 = 4.08
A study was conducted on the relationship between the speed of different different cars and their gasoline mileage, the correlation coefficient was to be 0.35. Later it was discovered that this defective speedometer was given values 5 mph too fast. What is the new correlation coefficient?
A study was conducted on the relationship between the speed of different different cars and their gasoline mileage, the correlation coefficient was to be 0.35. Later it was discovered that this defective speedometer was given values 5 mph too fast. What is the new correlation coefficient? Answer= 0.35 0.35 ---> 0.30 Positive r-correlation value indicates upward slope. If the the mileage readings wee 5mph too fast, the r-value wold remain the same. Since the sign of r tracks the slope of the overstimated line, then the actual correct r-correlation value would be negative
ANOVA Table Materials: SS= 900; DF= 2; MS=? Machines: SS= 2100; DF= 2; MS=? Errors: SS= 300; DF= 10; MS=? What are the missing MS values?
ANOVA Table Materials: SS= 900; DF= 2; MS=? Machines: SS= 2100; DF= 2; MS=? Errors: SS= 300; DF= 10; MS=? What are the missing MS values? SS / DF 900/ 2= 450, 2100/2 = 1050 300/ 10= 30 Answer: 450, 1050, 30
ANOVA and F-distribution
ANOVA and F-distribution ANOVA is a technique to determine if there is a significant difference in the mean of several treatments. F-distribution is used to determine significance and it is based on the assumption that the variance is the same for all treatments. If there are k treatments, and N observations for each treatment, the degrees of freedom are: DF for treatment k-1 DF of error k (n-1) total degrees of freedom nk-1
All multi-var i charts would initially plot a measurement for which of the category of variation?
All multi-var i charts would initially plot a measurement for which of the category of variation? Positional
An electronic components was made in quantities of 1M per year. Last year, 6 defectives were found. Plant manager asked master black belt to run 100,000 unit trial to determine with 95% confidence if the rate had been lowered by 2 DPMO. What would the black belt response be?
An electronic components was made in quantities of 1M per year. Last year, 6 defectives were found. Plant manager asked master black belt to run 100,000 unit trial to determine with 95% confidence if the rate had been lowered by 2 DPMO. What would the black belt response be? To prove that improvement of 33% had occurred, more than a year's production must be evaluated. Author calculates that 5-6 million must be examined. Increasing the interval of interest or lowering the confidence interval will help very little, bu tmillions of units will still be needed. Rough calculation gives 5.76 million units must be sampled. Section IX-31
An engineer has conducted a year-end analysis (365 data point) of incoming materials by checking for dimensional variation. For one of the items (a rule), the print dimension of length is to be 12.50 inches +/- 0.02 inches. the computer software indicated that the grand average was 12.52 inches and was statistically different from 12.5. What should engineer do? a. since the results is statistically significant, request the supplier to modify his equipment by 0.02 inches. b. follow the decision of the computer analysis, start rejecting the rules c. result is not of practical significance; the difference is too small to justify a change. d. supplier is to be notified via corrective action request to correct his process
An engineer has conducted a year-end analysis (365 data point) of incoming materials by checking for dimensional variation. For one of the items (a rule), the print dimension of length is to be 12.50 inches +/- 0.02 inches. the computer software indicated that the grand average was 12.52 inches and was statistically different from 12.5. What should engineer do? a. since the results is statistically significant, request the supplier to modify his equipment by 0.02 inches. b. follow the decision of the computer analysis, start rejecting the rules c. result is not of practical significance; the difference is too small to justify a change. d. supplier is to be notified via corrective action request to correct his process C= correct Practical significant is applies some though to the decision making process. Even though, a result is statistically significant, it may be of value to follow through
Basic assumptions underlying the analysis of variance?
Basic assumptions underlying the analysis of variance? Observations are normally distributed populations with equal variances.
Components of variance can be developed for a one-way ANOVA and will be based on a randomized model. Components of variance will enable the experimenter to determine which of the following: a. extent of contribution by each source of variance b. three component of variation c. maximum sum of squared errors d. how many additional tests will be required
Components of variance can be developed for a one-way ANOVA and will be based on a randomized model. Components of variance will enable the experimenter to determine which of the following: a. extent of contribution by each source of variance b. three component of variation c. maximum sum of squared errors d. how many additional tests will be required A=correct; ANOVA table does not require a COV calculation. However, a COV calculation can determine (A) (D) may or may not be predicratable with single experiment test. (B) describes a 2-way ANOVA (C)= one should determine the MINIMUM (not max) sum of squares
Consider the following multivari chart of single product measured in the same four locations, across width, over time.
Consider the following multivari chart of single product measured in the same four locations, across width, over time. Evaluating the chart by eye, arrange the categories of variation from largest to smallest. Temporal, positional , cyclical. During the plotted time interval, the greatest variation comes from temporal casues. Positional variation appears to be significant candidate for investigation. Cyclinci
A Tool that considers the joint variation of 2 measurements, neither of which is restricted by the experimenter
Correlation Analysis
Interdependence between 2 variables
Correlation Analysis Interdependence between 2 variables
Correlation and regression output between an input variable and output variable, returned the following values: slope = -3.27; intercept = -1.59; p-value = 0.18 At 90% confidence level, one can conclude: a. cannot conclude there is a correlation between the variables b. concludes there is negative correlation c. conclude the regression line has bias and zero offset d. conclude that there is an error in the data since both the slope and intercept are negative.
Correlation and regression output between an input variable and output variable, returned the following values: slope = -3.27; intercept = -1.59; p-value = 0.18 At 90% confidence level, one can conclude: a. cannot conclude there is a correlation between the variables b. concludes there is negative correlation c. conclude the regression line has bias and zero offset d. conclude that there is an error in the data since both the slope and intercept are negative. A= correct. 90% confidence level is equal to alpha= 0.10. Since the p-vale (0.18 ) is greater than the alpha, one cannot reject the H= and cannot conclude there is correlation between the two variables at a 90% confidence interval
Correlation coefficient and Coefficient of determination
Correlation coefficient and Coefficient of determination Correlation coefficient = r --a measure of correlation Coefficient of determination = r-squared If the Correlation coefficient is negative, then it is possible for the Correlation coefficient = r to be less than Coefficient of determination
Cotton supplier states that Mean breaking strength is 90 lbs. Random sample of 16....Standard deviation is unknown and normal curve theory instead of t distribution theory used to test the hypothesis. What type of error can occur?
Cotton supplier states that Mean breaking strength is 90 lbs. Random sample of 16....Standard deviation is unknown and normal curve theory instead of t distribution theory used to test the hypothesis. What type of error can occur? Type I = reject H0 Type II = accept H0 If alpha = 0.05, critical Z-value for type I risk (two-tail) = 1.960. Critical t-value for alpha (Type I risk, two tails) and DF= 15 is 2.131. When the normal curve Z-value is substitued for the proper t-value, there will greater risk of type I error SEction IX--24 and 37/41
Current process produces 500 lbs per shift. New processes produced 600 lbs per shift for 10 consecutive shifts. Highest shift was 660lbs and lowest shift was 540 lbs. Assuming normal distribution, what level of confidence can one say the process has changed?
Current process produces 500 lbs per shift. New processes produced 600 lbs per shift for 10 consecutive shifts. Highest shift was 660lbs and lowest shift was 540 lbs. Assuming normal distribution, what level of confidence can one say the process has changed? Answer is >99% Low value subtracted from the high value= 600 - 540 = 120 lbs= six sigma so the standard deviation= 120 / 6 = 20 t= X-bar - mean / S / square root of sample size t = 600 - 500/ 20/ Square root of 10 = 100 / 6.32456 = 15.81 Critical t-value for n-1 and 99% confidence is 3.250.
Decision that reliability test results show poor reliability, when in fact the reliability is acceptable is called?
Decision that reliability test results show poor reliability, when in fact the reliability is acceptable is called? Good (1-alpha) Decision
Determine the 95% confidence interval for a population proportion if 6 defectives were found in a sample size of 100 units
Determine the 95% confidence interval for a population proportion if 6 defectives were found in a sample size of 100 units Answer: 0.0135 < or equal to p < or equal to 0.1065 p +/- Z (Square root of [p (1-p)/ n] 0.06 +/- 1.96 (square root [0.06 *0.94/100]
Determine whether the following two types of rockets have significantly different variances at the 5% level. Assume that the larger variance goes in the numerator. Rocket A-- 61 readings ; 1,347 miles squared Rocket B-- 1,347readings ; 2,237 miles squared a. Significant difference because Fcalc < F table b. No Significant difference because Fcalc < F table c. Significant difference because Fcalc > F table d. No Significant difference because Fcalc > F table
Determine whether the following two types of rockets have significantly different variances at the 5% level. Assume that the larger variance goes in the numerator. Rocket A-- 61 readings ; 1,347 miles squared Rocket B-- 1,347readings ; 2,237 miles squared a. Significant difference because Fcalc < F table b. No Significant difference because Fcalc < F table c. Significant difference because Fcalc > F table d. No Significant difference because Fcalc > F table B= correct H0: There is a no difference int he variance between two groups Two-tailed test DF(1) = 30 DF(2) = 60 upper critical value of F, when alpha/ 2=0.0025, is 1.82 Variation = (standard deviation ) ^2 F= Variance (1) / Variance (2) F= (2,237 miles) / 1,347 miles) = 1.66 The question states that Variance (1) is larger variance. This avoids the tricky problem of finding the left-tail test. Since F calculated is the less than F-critical, the Ho: cannot be rejected. Our conclusion is that data only permits us to act as if the null hypothesis were true. One must say there is no significant difference. If the student wishes to determine the left-tail critical value, the degrees of freedom must be revered in the F=table and the reciprocal. That is, 1/ 1.94= 0.515. In this case, F -calculated is between the two critical value. Null hypothesis still cannot be rejected but the answer choices would be apply
Duplicate readings, using two separate instruments, were obtained from 10 samples. What is the tabular value of the statistic used to determine the significance of the mean difference at the 0.05 level of probability?
Duplicate readings, using two separate instruments, were obtained from 10 samples. What is the tabular value of the statistic used to determine the significance of the mean difference at the 0.05 level of probability? Answer = 2.26 Wording suggest that the critical value for paired t-test must ge determined. DF (t) = N-1= 10=1=9 Critical value (0.025, 9 ) = 2.26
What is extrapolation ?
Engineering judgement
Following Thread measurements: 4.06 3.88 3.87 3.97 4.03 3.97 3.91 4.10 4.06 3.98 What is the highest value that the machine output average may be, to 95% confidence?
Following Thread measurements: 4.06 3.88 3.87 3.97 4.03 3.97 3.91 4.10 4.06 3.98 What is the highest value that the machine output average may be, to 95% confidence? This is a t-distribution problem with X-bar= 0.079449, and 9 degrees of freedom. The t-value at 95% (= 0.05) is 1.833. Note--the value is to be greater than sample average mean = X-bar _ t (alpha) ( S/ square root of N) mean= 3.983 + 1.833 (o.079449 / Square root of 10) Mean = 3.983 + 0.04605 therefore mean= 4.029
For a linear correlation, the total sum of squares equals 1600, and the total sum of errors = 1000 What is R-square?
For a linear correlation, the total sum of squares equals 1600, and the total sum of errors = 1000 What is R-square? Answer = 0.375 R-square = SST- SSE / SST = 1600 -1000 / 1000 600/ 1600 = 0.375
From economic statistical staindpoint, the sample size of the hypotheiss testing would depend on ?
From economic statistical staindpoint, the sample size of the hypotheiss testing would depend on ? cost of samplesing
Given that random samples of process A produced 10 defective and 30 good units, while process B produced 25 defectives out of 60 units. Using chi-square test, what is the probability that the observed value of chi-square could result under the hypothesis that both processes are operating at the same quality level?
Given that random samples of process A produced 10 defective and 30 good units, while process B produced 25 defectives out of 60 units. Using chi-square test, what is the probability that the observed value of chi-square could result under the hypothesis that both processes are operating at the same quality level? Answer= between 5 % and 10% Chi Square Formula X-squared = Σ (0 - E) ^2 / E where O= observed and E= expected DF(chi-square) = (R-1) (c-1) Completed Chi-square table Process A Process B Total Defective : 10 (14) 25 (21) 35 Good: 30 (26) 35 (39) 65 Total 40 60 100 Expected Value = [ row total * column total] / Grand total X-square = [(10 -14) ^2] / 14 + (25-21) ^2 / 21 + (30-26)^2/26 + (35-39)^2 / 39 = 2.93 X-square (0.05) = 3.84 X-square (0.10) = 2.71
Given the data below, what is the 90% confidence interval for the variance? 22, 23, 19, 17, 29, 25
Given the data below, what is the 90% confidence interval for the variance? 22, 23, 19, 17, 29, 25 Answer = 8.27 ----79.88 Expression below is used to construct a confidence inerval for the variance of a sample taken from a normally distributed population, where n is the sample size and s is the sample standard deviation . Chi-square values have (n-1) degrees of freedom (n-1 ) * (Standard deviation ^2) / X^2 < Variance < (n-1 ) * (Standard deviation ^2) / X^2 Standard deviation = 4.278, n =6, the chi-square value is 0.05 and 5 degrees of freedom is 11.07 and the chi-square value at 0.95 and the 5 degrees of freedom is 1.1455 giving a confidence interval of 8.27 ----79.88
Given the population standard deviation is 6.8, what sample size is required to be 90% confident that the estimated mean has an error less than 0.02?
Given the population standard deviation is 6.8, what sample size is requied to be 90% confident that the estimated mean has an error less than 0.02? N= Z-squared * Sigma squared / E-squared N= (2.645 ^2) (6.8 ^2) / (0.02^2) = 312l816
H0: process has not improved as a result of some modification. Define type II errors.
H0: process has not improved as a result of some modification. Define type II errors ? Type II error means that one has failed to reject the Ho when it was false.
How is Sum of squares found in ANOVA Test?
How is Sum of squares found in ANOVA Test? SS= SST + SSE Therefore, SSE= Total SS - SST
How is r^2 affected as the scatter of points around the regression line becomes greater?
How is r^2 affected as the scatter of points around the regression line becomes greater? r-squared becomes smaller as the satter of points gets larger. r-squared = (Syy - SSE) / Syy
How many degrees of freedom for error variance if there are 3 machines? A 4 8 5 7 6 B 2 0 1 -2 4 C -3 1 -2 -1 0
How many degrees of freedom for error variance if there are 3 machines? A 4 8 5 7 6 B 2 0 1 -2 4 C -3 1 -2 -1 0 N= 15 samples t= 3 machines DF Error = N-t = 12
If 4 inspectors were evaluated, for the detection or non-detection of a defect in 20 samples, how many degrees of freedom would be used to determine the critical chi-square value?
If 4 inspectors were evaluated, for the detection or non-detection of a defect in 20 samples, how many degrees of freedom would be used to determine the critical chi-square value? DF= 3
If 90% confidence for the mean is 181.3 to 203.8, which is correct? a. 90% of all values in the population fall between 181.3 and 203.8 b. 90% of all values are greater than 203.8 or less than 181.3 c. Probability of randomly selecting a value between 181.3 and 203.8 is 90% d. None of above
If 90% confidence for the mean is 181.3 to 203.8, which is correct? a. 90% of all values in the population fall between 181.3 and 203.8 b. 90% of all values are greater than 203.8 or less than 181.3 c. Probability of randomly selecting a value between 181.3 and 203.8 is 90% d. None of above D= correct 90% confidence interval means that given the sample data, there is 90% chance that the true population mean is contained in the interval. AS the sample size increases, the width of the interval decreases.
If a sample size of 16 yields an average of 12 with standard deviation of 3, estimate the 95% confidence interval for the population (assume a normal distribution)
If a sample size of 16 yields an average of 12 with standard deviation of 3, estimate the 95% confidence interval for the population (assume a normal distribution) mu = (X-bar) +/- t (a/2) * (S/ square root on n) mu = 12 +/- 2.131 (3 / (square root of 16) = 12 +/- 1.6 10.4 < / = mean </ = 13.6
If multi-var chart shows 60% of variation is within piece, 25% of variation is piece-to-piece and 10% variation occurs over time. What would be the indicated improvement action sequence?
If multi-var chart shows 60% of variation is within piece, 25% of variation is piece-to-piece and 10% variation occurs over time. What would be the indicated improvement action sequence? Positional, Cyclical, Temporal
If the 95% confidence limits for mean m turn out to be 6.5, 8.5: a. probability is 0.95 that X-bar between 6.5 and 8.5 b. probability is 0.95 that X falls between 6.5 ad 8.5 c. probability is 0.95 that the interval (6.5, 8.5) contains mu d. 4sigma= 8.5 - 6.5
If the 95% confidence limits for mean m turn out to be 6.5, 8.5: a. probability is 0.95 that X-bar between 6.5 and 8.5 b. probability is 0.95 that X falls between 6.5 ad 8.5 c. probability is 0.95 that the interval (6.5, 8.5) contains mu d. 4sigma= 8.5 - 6.5 C= correct. For the population distribution, 3.92s, not 4s= 8.5 - 6.5. The 95% confidence interval is for m .......not X or X-bar.
If the coefficient of determination is 0.9, then what is the correlation of coefficient?
If the coefficient of determination is 0.9, then what is the correlation of coefficient? Answer= + / - 0.9487 If data is perfectly, positively correlated, r= 1 If data is perfectly negatively correlated, r = -1 Coefficient of determination = (correlation coefficent squared) The square root of 0.9 = + / - 0.9487
In a simple 2 variable linear regression study, what is the term B1 represent?
In a simple 2 variable linear regression study, what is the term B1 represent? Answer= slope of the line **B0 is the y -intercept
In an experiment designed to compare 2 different ways of measuring a given quality, it was desired to test the null hypothesis that the means were equal at the 0.05 level of significance. Method 1 --N= 5 Method 2 ---N= 7 t-ratio= 2.17
In an experiment designed to compare 2 different ways of measuring a given quality, it was desired to test the null hypothesis that the means were equal at the 0.05 level of significance. Method 1 --N= 5 Method 2 ---N= 7 t-ratio= 2.17 Question asking for finding the DF and critical value for a 2 mean t-test. Since no information given about standard deviation, assume that S1 = S2 DF = n1 + n2 -2 DF= 10 Critical value for t (0025, 10) = 2.228, the calculated 2.179 So Null hypothesis cannot be rejected.
In random sample of 900 vehicles, 80% had anti-lock brakes. What is the 95% CI for the percent of vehicles with anti-lock brakes?
In random sample of 900 vehicles, 80% had anti-lock brakes. What is the 95% CI for the percent of vehicles with anti-lock brakes? S= 900 80% of 900= 720 Confidence interval for proportions: p +/- Z square rool [p* (1-p) / n] z= 1.96, p= 0.8, and n= 900 giving confidence interval of 0.774-0.926
In single -factor ANOVA, what is the fundamental assumption that is made?
In single -factor ANOVA, what is the fundamental assumption that is made? Column variances are equal
In single factor, ANOVA the assumption of homogeneity of variance applies ?
In single factor, ANOVA the assumption of homogeneity of variance applies ? Variances within treatment groups is the same
In the broadest sense, how many of the following areas of variation in multi-vari analysis can include process time related elements?
In the broadest sense, how many of the following areas of variation in multi-vari analysis can include process time related elements? Positional, cyclical, and temporal. Piece-to-piece or batch-to-batch (cyclical) can reflect time. Positional measurements can reflect process time.
In the regression equation y= mx + b, y increases with x in all cases: a. if b is positive b. if b is negative c. if m is positive d. if m is negative
In the regression equation y= mx + b, y increases with x in all cases: a. if b is positive b. if b is negative c. if m is positive d. if m is negative C= correct If m> 0, y will increase as X increase. The intercept b can be either negative or positive.
In what statistical test is the two subtracted from the total number of samples to determine degrees of freedom?
In what statistical test is the two subtracted from the total number of samples to determine degrees of freedom? Two mean, equal variance, t-test
Key Fundamental ideas of ANOVA, SSE, and Means?
Key Fundamental ideas of ANOVA, SSE, and Means? Total Sum of Squares of Deviations from the gran means is equal to the sum of squares of deviations among treatment means and the gran mean plus th esum of squares of deviation within treatments. Total SS= SST + SSE
Define least squares method
Least squares method Definition = statistical procedure of finding the "best fitting " regression line. It , in many respect, a formalization of the procedure used when fitting a line by eye.
How is the level of significance defined?
Level of Significance (alpha risk) : Probability of rejecting the null hypothesis when it is true ----also called producer's risk.
Multi-vari chart indicates: 15% within piece variation 65% piece-to-piece variation 10% over time variation What is the recommended action sequence to reduce variation?
Multi-vari chart indicates: 15% within piece variation 65% piece-to-piece variation 10% over time variation What is the recommended action sequence to reduce variation? Cyclical, positional, temporal
New computer system requires you report the error rate within 0.5%, at a 95% confidence level. What sample size do you need if the population standard deviation is 1.2%?
New computer system requires you report the error rate within 0.5%, at a 95% confidence level. What sample size do you need if the population standard deviation is 1.2%? N= (Z^2 ) (Sigma squared) -------------------------------- E-squared N= (1.96) ^2 ( 0.12)^2 / (0.005)^2 N- 22.13 ---> 23 IX-31
No changes exists between a test trial and a current process. A sample size, that was too small for the change (delta) of interest, was then collect. What type of decision error would most probably be made?
No changes exists between a test trial and a current process. A sample size, that was too small for the change (delta) of interest, was then collect. What type of decision would most probably be made? Answer= 1- alpha decision In this situation, the null hypothesis would fail to be rejected. This is called a 1-slpha correct decision. Even if the sample were sufficiently large, the 1-alpha correct decision would be made.
Of the various statistical analysis tools available, which one would most likely to show a plot of all readings taken? a. X-bar-R-chart b. Multip-care charts c. ANOVA d. Chi-Square
Of the various statistical analysis tools available, which one would most likely to show a plot of all readings taken? a. X-bar-R-chart b. Multip-care charts c. ANOVA d. Chi-Square B= correct Only X-bar-R chart and multi-vari charts are plotted. The X-bar- R charts average data and ranges. Multi-vari chart normally contain all of the readings taken.
One-Way ANOVA Between: DF= 3 SSE= 55 Within: DF=15; SSE= 450.50 Total: DF-18; SSE= 505.50 How many runners were in the study?
One-Way ANOVA Between: DF= 3 SSE= 55 Within: DF=15; SSE= 450.50 Total: DF-18; SSE= 505.50 How many runners were in the study? Answer= 19; Total Df= 18 n-1= 18
One-way ANOVA table bewlo. What are the MS value Source df Sum of Squares MS Between 3 55.00 Within 15 450.00 Total 18 505.00
One-way ANOVA table bewlo. What are the MS value Source df Sum of Squares MS Between 3 55.00 Within 15 450.00 Total 18 505.00 Answer= (18.3 , 30.0) "Between" refers to additives and describes the variation present among the runners. Mean Square = sum of Squares / Degrees of Freedom 55/ 3 = 18.3 450/ 15 = 30
Opposite of alpha risk in hypothesis testing is called?
Opposite of alpha risk in hypothesis testing is called? Level of Confidence or (1- alpha) ......Not Beta risk
Product was yielding 90% recover before the improvement. To determine if a 2% changes ( in either direction ) would be made at the 95% confidence interval, what sample size should be taken?
Product was yielding 90% recover before the improvement. To determine if a 2% changes ( in either direction ) would be made at the 95% confidence interval, what sample size should be taken? Use binomial formula: n= (Z-squared) (p-bar) ( 1- pbar) / (Change in P) ^2 N= (1.96^2) ( 0.90) (0.10) / (0.02^2) = 864
Random sample size is to be taken from a large population having a standard deviation of 1 inch. The sample size is to be determined so that there will be 0.05 risk probability of exceeding a 0.1 inch tolerance eror in using the sample mean to estimate "mu". Which of the following values is required sample size?
Random sample size is to be taken from a large population having a standard deviation of 1 inch. The sample size is to be determined so that there will be 0.05 risk probability of exceeding a 0.1 inch tolerance eror in using the sample mean to estimate "mu". Which of the following values is required sample size? Correct answer= 385 n= [ (z^2) (sigma ^2) ] / E^2 **E= desired interval; z= confidence interval z-value n= (1.96 ^2) (sigma ^2) / E^2 n= (1.96 ^2) (1 ^2) / (0.1^2) = 384.16
Randomly selected ample of helmets tested. What is the 95% confidence interval? N= 100 Mean= 276 Standard deviation= 15
Randomly selected ample of helmets tested. What is the 95% confidence interval? N= 100 Mean= 276 Standard deviation= 15 276 +/- 2.94 E= 1.96 ( 15 / square root of 100) E= +/- 2.94
Mathematical expression that describes the relationship between two or more variable, normally to make a projection into an untested region.
Regression Analysis Mathematical expression that describes the relationship between two or more variable, normally to make a projection into an untested region.
Relation of t-distribution, DF, and critical value
Relation of t-distribution, DF, and critical value Critical value of t-distribution approaches the critical value of the normal distribution as the degrees of freedom increase
Result from a fixed factor, randomized, block design experiment in which the production outputs of 4 machines (A, B, C, D) are compared. A 4 8 5 7 6 B 2 0 1 -2 4 C -3 1 -2 -1 0 D -5 -4 +2 +1 =2 How many degrees of freedom are used to compute the error variance? A. 4 B 12 C. 16 D. 19
Result from a fixed factor, randomized, block design experiment in which the production outputs of 4 machines (A, B, C, D) are compared. How many degrees of freedom are used to compute the error variance? A. 4 B 12 C. 16 D. 19 Equations: DF total= n-1 ----> Block has total of 20 numbers or results so----20-1 = 19 DF treatments = t- 1 ----> 4 -1 = 3 DF error = n- t ----> 20 -4 = 16 where n= sample size and t= number of machines
Results for one-way ANOVA. What is the critical F-value for the "Between" Sources, and significance of this factro. P < 0.05 SSE= sum of squares Between df=3; SSE=55 MS=? Within: df=14; SSE=450 MS=? Total: df=18; SSE=505; MS=?
Results for one-way ANOVA. What is the critical F-value for the "Between" Sources, and significance of this factro. P < 0.05 SSE= sum of squares Between df=3; SSE=55 MS=? Within: df=14; SSE=450 MS=? Total: df=18; SSE=505; MS=? Answer= Not significant (3.29. 0.61) Between= additives; Within= error; describes the variation among units. F=table at a value of 0.05, using 3 df, and 15 df is 3.29 F= ( Between MS) / (Within MS) = (55/ 450) = 0.61 No significance because the F (table value) =3.29 is greater than the the F (Calculated value) = 0.61
Six independent samples were taken from a smoke stack. Mean value of the critical pollutant was 30.2 ppm for the six samples, with standard deviation S= 5ppm. Assuming random sampling, what is the 90% confidence interval for the proportion of the critical pollutant?
Six independent samples were taken from a smoke stack. Mean value of the critical pollutant was 30.2 ppm for the six samples, with standard deviation S= 5ppm. Assuming random sampling, what is the 90% confidence interval for the proportion of the critical pollutant? Degrees of Freedom = N-1 = 5, alpha = 0.10 From t-table, t (a/2) = 2.015 E= +/- t (a/2) * S/ Square root of N E= +/- 2.015 (5 / square root of 6) = +/- 4. 11
Small positive change truly exists between the test trial and current process. However, the sample from the test trial happens to demonstrate a radical improvement. What type of decision would probably be made?
Small positive change truly exists between the test trial and current process. However, the sample from the test trial happens to demonstrate a radical improvement. What type of decision would probably be made? Answer= 1-beta decision---- correct decision has been made for the wrong reason. A false null hypothesis was rejected for the wrong reason IX-25
Suppose that , given X-bar = 35, and Z (0.05)= +/- 2.58, one established the confidence limits for mean of 30 and 40. this means?
Suppose that , given X-bar = 35, and Z (0.05)= +/- 2.58, one established the confidence limits for mean of 30 and 40. this means? Answer= the probability that the interval contains the mean is 0.99.
To determine the difference between treatment means, there is a break point between using an analysis of variance or t-test. When can a t-test be used?
T-test can be used to determine the difference between means when there are 2 or more means.
Tennis racket string tension measurements: 57 lbs, 56 lbs, 60 lbs, 62 lbs, 55 lbs, and 59 lbs What are the 90% confidence limits for the estimated true value of string tension
Tennis racket string tension measurements: 57 lbs, 56 lbs, 60 lbs, 62 lbs, 55 lbs, and 59 lbs What are the 90% confidence limits for the estimated true value of string tension Mean= 58.17 Standard deviation = 2.64 N= 6 DF= 5, alpha= 0.10 T-table ---> t (a/2) = 2.015 E= +/- 2.015 * (2.64 / square root of 6) E = +/- 2.17 lbs.
Test of significance using a given value of "a" is performed on the yield data from a process, using a standard material and a proposed substitute. Which of the following conclusions is possible from this test? a. the standard material is better than the substitute material. b. there is an interaction between the two materials. c. Probability of type I error is beta. d. sample size is too large to detect any material differences.
Test of significance using a given value of "a" is performed on the yield data from a process, using a standard material and a proposed substitute. Which of the following conclusions is possible from this test? a. the standard material is better than the substitute material. b. there is an interaction between the two materials. c. Probability of type I error is beta. d. sample size is too large to detect any material differences. A= correct (a) - part of the failure to reject a null hypothesis is the implication that insufficient sample size may exist (b) is false. (d) --A part of the failure to reject a null hypothesis is the implication that an insufficient sample size may exist. Quite often, a determination of the proper sample size can be made. (c) is wrong, type I error is the alpha risk.
The coefficients in the equation below can be determined using Yi = a + bx(i) + e (i) A. Inference testing b. Least squares regression c. Sum of Squares estimation d. Hypothesis testing
The coefficients in the equation below can be determined using Yi = a + bx(i) + e (i) A. Inference testing b. Least squares regression c. Sum of Squares estimation d. Hypothesis testing B= correct; The equation above represnts a straight line a= y-intercept ; b= slope; e= error term B= correct; Least Squares Method ---> used to determine the coefficients of a andb
The difference between setting alpha equal to 0.05 and alpha equal to 0.01 in hypothesis is?
The difference between setting alpha equal to 0.05 and alpha equal to 0.01 in hypothesis is? With alpha equal to 0.05 , one is will to risk a type I error An alpha or type I risk is the risk of rejecting a true hypothesis. When moving from alpha equal to 0.01 to 0.05 ( 99% assurance to 95% assurance), one is more willing to accept a type I error.
To attract business travelers, an airline places a faster aircraft on a demanding, competitive route to St. Louis to Atlanta. Weather and air traffic are significant factors along the route, so the company wnats to be sure that the faster aircraft is making the trip in less time. With the previous aircraft, the airborne time averaged 74 minutes with standard deviation of 5.2 minutes. the first five flights with faster aircraft resulted in airborne times of 70, 82, 62,65,74, and 68 minutes. What are the critical and calculated test values for 95% reliability, and did the airborne time decrease?
To attract business travelers, an airline places a faster aircraft on a demanding, competitive route to St. Louis to Atlanta. Weather and air traffic are significant factors along the route, so the company wnats to be sure that the faster aircraft is making the trip in less time. With the previous aircraft, the airborne time averaged 74 minutes with standard deviation of 5.2 minutes. the first five flights with faster aircraft resulted in airborne times of 70, 82, 62,65,74, and 68 minutes. What are the critical and calculated test values for 95% reliability, and did the airborne time decrease? Answer= Critical = -2.132, Calculated= -301, reject the Ho--flight time did increase. Statistical inference problem. Ho: new flight time is not less, or that the difference between the flight time means does not provide 95% confidence that the time is less. New average time= X-bar= 67.8 Standard deviation = 4.6 Since the calculated value falls outside the critical value, reject the hypothesis. Ho: Mean (2) is greater than or equal to Mean (1) DF= n-1= 4; alpha= 0.05 t (critical) = -2.132 (from table) t (calculated) = (X-bar - mean) / [standard deviation/ square root of sample] t (calculated) = 67.8 -74/ [4.6 / square root of 5] = -3.01
Two samples were taken from the production shown. What is the F-value, for the usual test of equality, to compare the variances of the two distrubitons? Sample 1 Sample 2 16.10 16.15 16.13 16.20 16.17 16.16 16.10 16.11 16.17 16.15 16.18 16.13 16.14 16.15 16.08 16.13 16.15 16.23 16.14 16.16
Two samples were taken from the production shown. What is the F-value, for the usual test of equality, to compare the variances of the two distrubitons? Sample 1 Sample 2 16.10 16.15 16.13 16.20 16.17 16.16 16.10 16.11 16.17 16.15 16.18 16.13 16.14 16.15 16.08 16.13 16.15 16.23 16.14 16.16 F-value= larger value variance / smaller value variance. Both samples have DF= 9 Sample 1 variance = 0.001138 Sample 2 variance = 0.001223
Two-way analysis of variance has r levels for one variable and c-levels fro the second variable with 2 observations per cell. Degrees of freedom for interaction is?
Two-way analysis of variance has r levels for one variable and c-levels fro the second variable with 2 observations per cell. Degrees of freedom for interaction is? Answer= (r-1) (c-1) DF in a 2-way ANOVA is always (r-1) (c-1) regardless of the number of interactions. Number of replications is used to determine the sum of squares, not DF.
Type II error definition
Type II error definition Ho is not rejected when it should be.
Using a one-way ANOVA implies what about the samples?
Using a one-way ANOVA implies what about the samples? Samples are independent
Valid reason for conducting multi-vari studies?
Valid reason for conducting multi-vari studies? They identify areas to direct improvement activities
What are the F-values for the partial ANOVA table: Materials : SS= 900; DF = 2; MS = 450 Machines : SS= 2100 ; DF= 2; MS= 1050 Errors SS= 300l DF= 10; MS = 30 Total DF= 14
What are the F-values for the partial ANOVA table: Materials : SS= 900; DF = 2; MS = 450 Machines : SS= 2100 ; DF= 2; MS= 1050 Errors SS= 300l DF= 10; MS = 30 Total DF= 14 Row Mean Square ________________ Error Mean Square Row Mean Square = 450 Error Mean Square= 30 450 / 30 = 15 1050/ 30 = 35 Asnwer= 15, 30
What does determination of temporal variation in multi-vari mean?
What does determination of temporal variation in multi-vari mean? variation over time; TEMPORAL key word
What inference test compares observed and expected frequencies of test outcomes?
What inference test compares observed and expected frequencies of test outcomes? Chi-square test
What is the basic assumption required in making a null Hypothesis?
What is the basic assumption required in making a null Hypothesis? Must assume that variables are independent
What is the major variation classification specifically addressed in mult-var studies?
What is the major variation classification specifically addressed in multi-var studies? Positional, cyclical, temporal
What is the missing critical values, assuming 95% confidence? Materials: SS = 900; DF=2; MS=450; F= 15; Critical value? Machines SS = 2100; DF=2; MS=1050; F= 35; Critical value? Errors: SS = 300; DF=10; MS=30; F= 15; Critical value? Total DF= 14
What is the missing critical values, assuming 95% confidence? Materials: SS = 900; DF=2; MS=450; F= 15; Critical value? Machines SS = 2100; DF=2; MS=1050; F= 35; Critical value? Errors: SS = 300; DF=10; MS=30; F= 15; Critical value? Total DF= 14 H0: All materials and machines are equal are rejected. F (o.05, 2, 10) = 4.10 F (o.05, 2, 10) = 4.10 Answer= 4.10, 4.10 See IX- 53
What is the z-valued needed to conduct a two-tail test in statistical inference problem, specifiying 90% confidence level?
What is the z-valued needed to conduct a two-tail test in statistical inference problem, specifiying 90% confidence level? Range= X-bar +/- t(0.025) (S/ square root N) Range = 21.07 +/- 2.262 (0.1259/ square root of 10) = 21.07 +/- 0.09 = 21.16 / 20.98
What is true if the coefficient of determination is 0.85?
What is true if the coefficient of determination is 0.85? 85% of the variability is explained by regression model
What would be the F-ratio value if the means of the samples fro each group were identical?
What would be the F-ratio value if the means of the samples fro each group were identical? F= (Mean variation between treatment) / Mean variation within treatments) F= MST / MSE = 0/MSE So F=0
When finding a confidence interval for mean "mu", based on sample size of n: a. Increasing n increases the interval b. Having to use S(x), instead of n decreases the interval c. the larger the interval, the better the estimate of mu d. Increasing the n decreases interval
When finding a confidence interval for mean "mu", based on sample size of n: a. Increasing n increases the interval b. Having to use S(x), instead of n decreases the interval c. the larger the interval, the better the estimate of mu d. Increasing the n decreases interval D= correct mean = 1.96 +/- sigma / (square root of n) ***The larger the interval, the poorer the estimate of m. From above equation, as the sample size is increased, the confidence interval will become smaller.
When making a chi-square determination where do the degrees of freedom come from?
When making a chi-square determination where do the degrees of freedom come from? Degree of Freedom are the product (rows -1) (columns -1)
When performing a linear regression analysis, it is not unusual to have identical t-statistics for which 2 variables? a. b0 and correlation coefficient b. b1 and coefficient of determination c. b1 and correlation coefficient d.b0 and y
When performing a linear regression analysis, it is not unusual to have identical t-statistics for which 2 variables? a. b0 and correlation coefficient b. b1 and coefficient of determination c. b1 and correlation coefficient d.b0 and y B= correct. ---b0 is y-intercept. b1 is the slope; correlation coefficient= indicates if a linear relationship exists Coifficient of determination= how well the regression line fits the data
When plotting a multi-var chart on graph paper, what metric is used for the horizontal scale?
When plotting a multi-var chart on graph paper, what metric is used for the horizontal scale? Horizontal scale= Time or sample number in time sequence Vertical Scale = variable measurements ( usually with locational identification)
Which input parameter has the smallest impact when performing statistical tests?
Which input parameter has the smallest impact when performing statistical tests? Population size that is not a "must" when discriminating differences in input parameters Population variance, risk level ( alpha and beta) , population variance, and parameter shift sensitiviy have a large impact when performing statistical tests
Which of the following best describes a p-value? a. similar to the critical statistical value found in the statistical table b. containing an extreme test statistic probability value as obtained from the sample data. c. having the typical set values of 5% to1 % d. being reported only when significant
Which of the following best describes a p-value? a. similar to the critical statistical value found in the statistical table b. containing an extreme test statistic probability value as obtained from the sample data. c. having the typical set values of 5% to1 % d. being reported only when significant B= correct. ---p-value is the probability of getting a value of the sample test statistic that is at least as extreme as the one found from the sample data (assuming that the hypothesized value is correct) (A) wrong b/c statistical table describe the critical values (C) wrong b/c set values of 5% to1 % are used to determine the boundaries of the significance of hypothesis
Which of the following confidence interval calculation requires the use of t-table values?
Which of the following confidence interval calculation requires the use of t-table values===Small Sample Means
Which of the following is a valid null hypothesis? a. p < 1/8 b. mu < 98 c. mean of population A is not equal to the mean of population B d. mu =110
Which of the following is a valid null hypothesis? a. p < 1/8 b. mu < 98 c. mean of population A is not equal to the mean of population B d. mu =110 D= correct Statistical hypothesis test determines if there is enough evidence to reject the null hypothesis at a given significance level. Null hypothesis must contain an equal sign. For example, when testing, if two populations have the same mean, the assumption is that the populations have equal means, and data is collected in a attempt to reject this hypothesis. If the H0 i s not accepted, it does not mean the means of the two populations are equal, but there is not sufficient to reject the null hypothesis that jockeys and basketball player have the same height. a much larger sample size would be required to reject the Ho that basket players in US are the same height as Europe's basket players . Unless the 2 populations are exactly the same. the Ho can be rejected if enough data is collected
Which of the following statements concerning statistical inference is true? a. Confidence interval is a range of values which does not include the true value of a population parameter. b. confidence interval is normally the statistical tolerance limits of the population parameter c. Point estimate is a single value used to estimate the population parameter d. estimation is the process of analyzing the population parameter in order to predict the value of a sample result.
Which of the following statements concerning statistical inference is true? a. b. confidence interval is normally the statistical tolerance limits of the population parameter c. Point estimate is a single value used to estimate the population parameter d. estimation is the process of analyzing the population parameter in order to predict the value of a sample result C= correct; (B) is false. Sample confidence interval may or may not align with the tolerance limits of the population parameter. That is why statistical inference tests are conducted. (A) does not make sense
Which of the following statements is true? a. confidence interval increase in width as the sample size increases b. confidence intervals are always symmetrical c. Confidence intervals for the mean are independent of the population distribution d. confidence intervals are independent of the sample size
Which of the following statements is true? a. confidence interval increase in width as the sample size increases b. confidence intervals are always symmetrical c. Confidence intervals for the mean are independent of the population distribution d. confidence intervals are independent of the sample size C= correct Confidence intervals for the mean are usually the only confidence intervals that are symmetrical. Other confidence intervals, such as those for the variance, are not.Confidence intervals decrease in width as the sample size increases.
Which statements about coeffiecient of correlation (r ) is true? a. r will never have the same sign as b1 b. r equals 0, when b1 =1 c. When r equals 1, b1 is positive d. r and b1 have no directional relationship
Which statements about coeffiecient of correlation (r ) is true? a. r will never have the same sign as b1 b. r equals 0, when b1 =1 c. When r equals 1, b1 is positive d. r and b1 have no directional relationship C= correct. (B) false.
Which table value is normally used to place a confidence interval on the slop of a line in simple linear regression?
Which table value is normally used to place a confidence interval on the slop of a line in simple linear regression? answer- t-value In most experiments, one works with small samples therefore used t-values
Which test statistic must be known in order to compute the confidence interval for variation?
Which test statistic must be known in order to compute the confidence interval for variation? Chi-square value IX-36 ; see the formula for confidence interval for variation.
Why is ANOVA used to test for significance?
Why is ANOVA used to test for significance? Equality of sample means can be tested by comparing sample variances. To determine if the "within" treatment variation is significant in comparision to treatment means
If no correlation exists between 2 variables, then: a. correlation coefficient should equal a negative value b. as one variable changes, one cannot predict a value for the other variable c. both variables will decrease simultaneously d. analysis of variance must be used to determine if an interaction is present.
f no correlation exists between 2 variables, then: a. correlation coefficient should equal a negative value b. as one variable changes, one cannot predict a value for the other variable c. both variables will decrease simultaneously d. analysis of variance must be used to determine if an interaction is present. B= correct.
