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b) A meat packaging department of a supermarket chain wished to evaluate the efficiency of their new vacuum sealing process. To do this they monitored the number of days it took for the packaged meat to spoil. The number of days follows a normal distribution with a standard deviation of 1.90 days. A sample of 60 packages gave a mean number of days to be 7.20. Calculate 90% and 97% confidence intervals for the mean number of days it takes for the meat to spoil.

b) A meat packaging department of a supermarket chain wished to evaluate the efficiency of their new vacuum sealing process. To do this they monitored the number of days it took for the packaged meat to spoil. The number of days follows a normal distribution with a standard deviation of 1.90 days. A sample of 60 packages gave a mean number of days to be 7.20. Calculate 90% and 97% confidence intervals for the mean number of days it takes for the meat to spoil.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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QUESTION 3 b- problem-type question requiring detailed written answers.

 

  • Table 3 below shows all the possible values of a discrete random variable, X, and their corresponding probabilities.  

 

X 

−3

−1 

0 

1 

3 

P(X = x) 

0.14 

0.22 

0.36 

0.20 

0.08 

 

 b) A meat packaging department of a supermarket chain wished to evaluate the efficiency of their new vacuum sealing process. To do this they monitored the number of days it took for the packaged meat to spoil. The number of days follows a normal distribution with a standard deviation of 1.90 days. A sample of 60 packages gave a mean number of days to be 7.20. Calculate 90% and 97% confidence intervals for the mean number of days it takes for the meat to spoil.

z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.5 1.6 1.7 1.8 20 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 5000 5398 5793 6179 3.2 3.3 6554 6915 7257 7580 7881 8159 8413 8643 8849 9032 9192 9332 9452 9554 9641 9713 9772 9821 9861 9893 9918 9938 9953 9965 9981 9987 00 -3.4 0003 0005 -3.2 0007 -3.1 0010 -3.0 -2.9 0019 -2.8 0026 -2.7 0035 0047 -2.5 0062 01 5040 5438 5832 6217 -14 0808 -1.3 0968 -1.2 1151 -1.1 1357 -1.0 1587 6591 6950 -0.9 1841 -0.8 2119 -0.7 2420 -0.6 2743 -0.5 7291 7611 7910 8186 8438 -0.4 3446 -0.3 3821 -0.2 4207 -0.1 0.01 5000 8665 8869 9049 9207 9345 9990 9993 9995 9997 This is Table IV of Appendix C. 9463 9564 9649 9719 9778 9826 9864 9896 9920 9940 9955 9966 9975 9982 9987 9991 9993 9995 9997 01 0003 0005 0007 0009 0013 -1.9 0287 0281 -1.8 0359 -1.7 0446 0018 0025 0034 0045 0060 -1.6 0548 0537 -1.5 0668 0655 0351 0436 0793 0951 1131 1335 1562 1814 2090 2389 2709 3050 3409 3783 102 4168 5080 4960 5478 5871 6255 6628 6985 7324 7642 7939 8212 8461 8686 8888 0082 0080 0078 0132 -2.3 0107 0104 0102 -2.2 0139 -21 -2.0 0228 0136 0174 0179 0170 0222 0217 9066 9222 9357 9474 9573 9656 9726 9783 9830 9868 9898 Table IV Standard Normal Distribution Table 9922 9941 9956 9967 9976 9982 9987 9991 9994 9995 9997 1002 0001 0005 0006 0009 0018 0024 0033 0044 0059 0274 0344 0427 0526 0643 0778 0934 1112 1314 1539 1788 2061 2358 2676 3372 3745 4522 4920 003 5120 5517 5910 6293 6664 7019 7357 7673 7967 8238 8485 8708 8907 9082 9236 9370 9484 9582 P(1) --- xC₂ Poisson probability formala P(x) 9834 9871 9901 9925 9943 9957 9968 9977 9495 9591 9664 9671 9732 9738 9788 9793 9983 9988 9991 9994 9996 9997 normal curve to the left of z with the values of z equal to 0 or positive. 03 0003 0004 0006 0009 0012 0017 0023 0032 0043 0057 0075 0099 0129 0166 0212 0268 0336 0418 0516 0630 0764 0918 1093 1292 1515 1762 2033 2327 3336 3707 4483 4880 Joint probability of two mutually exclusive events: PA and B) 0 . Addition rule for mutually nonexclusive events: P(AB)=P(A) + P(B)- PA and B) Addition rule for mutually exclusive events: P(AB)=P(A) + P(B) Chapter 5. Discrete Random Variables and Their Probability Distributions Mean of a discrete random variable x:=xP(x) . Standard deviation of a discrete random variable x: a-√2+¹P(1)-² 04 5160 5557 5948 6331 Chapter 6. Continuous Random Variables and the Normal Distribution 2 value for an x value Value of x when . o, and are known: x=+50 7054 Chapter 7. Sampling Distributions - Mean of : . Standard deviation of I when n/Ns .05 aya/V I-K 7389 7704 7995 8264 8508 8729 8925 9099 9251 factorial: w-1-2) 3-2-1 Number of combinations of a items selected x at a time: F.x-x) . Number of permutations of items selected x at a time: 04 0003 0004 0006 0008 0012 (-x) Binomial probability formulac P(x)-C, pq Mean and standard deviation of the binomial distribution: panday Hypergeometric probability formula: 9959 9969 9977 9984 0016 0023 0031 0041 0055 9992 9994 9996 9997 0073 0096 0125 xt . Mean, variance, and standard deviation of the Poisson prob ability distribution: =A²A, and VĀ 0207 0262 0329 0409 0505 0618 0749 0901 1075 1271 1492 2005 2296 2611 3300 3669 The entries in this table give the cumulative area under the standard normal curve to the left of z with the values of z equal to 0 or negative. 4443 9838 9842 9875 9878 9904 9906 9927 9945 4840 005 5199 5596 5987 6368 6736 7088 7422 7734 8023 8289 8531 9505 9599 9678 9744 9798 9929 9946 9960 9970 8749 8770 8944 8962 9115 9131 9265 9279 9394 9406 9984 9989 9992 9994 9996 9997 05 0003 0004 0006 0008 0011 0016 0022 0030 0040 0054 0071 0094 0122 0158 0202 0256 0322 0401 0495 0606 0735 0885 1056 1251 1469 1711 1977 2266 2578 2912 3264 3632 06 5239 4404 4801 56.36 6026 6406 6772 7123 7454 7764 8051 9515 9608 8315 8340 8554 8577 9686 9750 9803 9961 9971 9985 9992 9994 9996 9997 06 9846 9850 9881 9884 9909 9911 9931 9932 9945 9949 0003 0004 0006 000 0011 0015 0021 0039 0052 0091 0119 0197 0250 0314 0392 0485 0594 0721 0869 1038 1446 1685 1949 2236 2546 3228 3594 4364 1867 5279 3675 6064 6443 4761 7157 746 7794 8078 of 8980 9147 9292 9418 9525 9616 9693 9756 9808 9962 9972 9979 9985 9995 9996 9997 107 0003 0004 0005 0008 0011 0015 0021 0028 0038 0051 0089 0116 0150 0192 0244 0307 0384 0475 0582 0708 0853 1020 1423 3192 3557 108 08 5319 5714 6103 6480 7190 7517 7823 8106 8365 8599 - Margin of error of the estimate for a E20 of th - Determining sample size for estimating 8810 8997 9162 9306 9429 9535 9625 9854 9913 9934 9951 9963 9973 9956 0003 0004 0005 0007 9699 9706 9761 9767 9812 9993 9995 9996 0014 0020 where=1/√ñ 0037 0049 0066 0087 0113 0146 0188 0239 0301 0375 pt 2 where Margin of error of the estimate for p E-2 where - V Determining sample size for estimating p -²/E² 0465 0571 0694 0838 1003 1401 4681 5359 5753 6141 6517 6879 7549 8133 8389 8830 9015 Confidence interval for p for a large sample: V/n 9177 9319 9545 Chapter 9- Hypothesis Tests about the Mean and Proportion 9857 9890 9916 9936 9952 9964 9986 9990 9993 9995 9997 9998 09 0002 0003 0005 0007 0010 1660 1635 1922 1894 2206 2177 2148 2514 2483 2451 2843 2810 2776 0019 0048 0064 0084 3156 3520 3897 3859 4325 4286 4247 4721 4641 (continued on next page) 0110 0143 0183 0233 0294 0367 0455 0559 0681 0823 0985 1170 1379 - Population proportion: p-X/N Sample proportion: -x/ - Mean of p-p Standard deviation of when a/N05: Vpq/n 1611 1867 Chapter 8- Estimation of the Mean and Proportion - Point estimate of i .Confidence interval for using the normal distribution when is known 3121 3483 try where aya/V Confidence interval for a using the r distribution when or is Page 11 of 13 Test statistic z for a test of hypothesis about ja using the normal distribution when is know 2 where y Test statistic for a test of hypothesis about ja using the t dis tribution when or is not known -Test statistic for a test of hypothesis about p for a large sample: -- where - Page 9 of 13 M8 38333 88889 30212 9548A HAAaa aaaaa 3.1 3.3 00 3000 5398 3793 6179 6554 6915 7257 7580 8643 8849 9032 9192 9332 9452 9554 9641 9713 9772 7881 7910 81.59 8186 8413 8438 9861 9893 9918 9938 9953 9965 9974 01 9981 9987 3040 3438 5832 6217 6591 6950 7291 7611 3665 8869 9207 9345 9463 9564 9649 9719 9876 9864 9896 9920 9940 9955 9966 9975 9982 9987 9990 9993 9995 9995 9997 9997 This is Table IV of Appendix C. -- ww 9993 002 5080 Within-samples sum of squares 5478 5871 6255 6985 7324 7642 7939 8212 8461 Total sum of squares: 9726 9783 9830 Chapter 12- Analysis of Variance Let 9898 9922 9941 the mumber of different samples (orta) 9956 9967 9976 9982 9987 . Cumulative percentage 9991 9994 9995 8485 8686 8708 8888 81417 9066 9082 9222 9236 9357 9370 9474 9573 03 2²- .Confidence interval for the population variance o X- to Test statistic for a test of hypothesis about N Test statistic for a goodness-of-fit test and a test of inde- (0-1 5120 Zury- (Emyp 5517 6293 2004 the size of sample 7,- the sum of the values in sample i the number of values in all samples 7357 7673 7967 - ++ the sum of the values in all samples the sm of the squares of values in all samples For the F distribution and normal curve to the rest of z with the values of z equal to 0 or positive. 9484 9582 9664 9732 9788 Degrees of freedom for the somerator--1 Degrees of freedom for the denominator-k Between-samples sum of squares - Range - Largest value - Smallest value . Variance for ungrouped data (Ex)² IP- 9834 9871 9901 9925 SST SSB SSW-1 Variance between samples: MSB SSB/A-1) Variance within samples: MSW-SSW/(-A) Test statistic for a one-way ANOVA lest FMSB/MSW 9957 9968 9977 9983 99k 9991 9994 9996 9997 Chapter 13 Simple Linear Regression Simple linear regression model: y= A + B + e Estimated simple linear regression model: 9-a + bx Prem S. Mann Chapter 2- Organizing and Graphing Data . Relative frequency of a class f/2f N and ²- a² m - Standard deviation for grouped data: (Emy) Smy-N . Percentage of a class (Relative frequency) x 100 . Class midpoint or mark (Upper limit +Lower limit)/2 . Class width Upper boundary - Lower boundary . Cumulative relative frequency 04 - (Cumulative relative frequency) x 100 Chapter 3 - Numerical Descriptive Measures - Mean for ungrouped data: Ex/N and 7-x/m Mean for grouped data: Emf/N and I Σmf/n where is the midpoint and / is the frequency of a class Median for ungrouped data Value of the middle term in a ranked data set 2² (+)² 5100 5557 5948 6331 6700 7054 7389 7704 7995 8264 8508 8729 8925 Cumulative frequency Total observations in the data set 9251 9382 - 9495 9591 9671 9738 9793 and where is the population variance and is the sample Standard deviation for ungrouped data (Ex) 9838 9875 9904 9927 9945 In'y- (Imp)? 9959 9900 9992 9994 9996 9997 9984 9988 N and s #-1 where and s are the population and sample standard de viations, respectively Variance for grouped data Emp 05 5199 5596 3987 6368 6736 7088 -V Chebyshev's theorem: For any number & greater than 1, at least (1-1/2) of the values for any distribution lie within k standard deviations of the mean 7422 7734 8023 8289 8749 8944 9115 9265 9394 9505 9678 9744 9798 9842 9878 9906 9929 9946 06 3239 3636 6026 6772 7123 7454 7764 8051 8315 8554 8770 8962 9131 9279 9406 9515 9608 9686 9750 9803 9846 9881 9909 9931 9948 9961 9960 9970 9978 9984 9989 9989 9992 9994 9996 9997 9992 9994 9996 9997 Sum of squares of xy, x, and 55,-2-(X) 17 3279 3675 6064 6443 6808 7157 .Confidence interval for 92, where Prediction interval for 7486 7794 8078 8340 8577 KEY FORMULAS Introductory Statistics, Sixth Edition 8790 3980 9962 9972 9979 9979 9985 9985 9989 9292 9525 9616 9693 9756 9884 9911 9932 9949 9992 9995 9996 9997 3319 5714 6103 6430 08 6841 7190 7517 7823 8106 8365 8599 8810 www 9162 9306 9429 9533 960 9761 9812 9887 9913 9934 9951 Least squares estimates of A and B -55/55, and any-A Standard deviation of the sample en 9961 9973 9980 9986 9990 www. 59995 9996 9997 Error sum of squares SSE-1-2(y-99 (₂) Total sum of square SST-1²- Regression sun of squees SSRSST-SSE . Coefficient of determination: -55/55 .Confidence interval for B bt, where 44/V55, Test statistic for a test of hypothesis about B Linear correlation coefficient VSS, 55 Test statistic for a test of hypothesis about po (₂-19 99 5359 5753 6141 6517 Interquartile range: IQR-0-0 - The Ath percentile: 7224 7549 7852 8133 8621 9015 9177 9319 9545 9706 9767 9817 Chapter 4- Probability Classical probability rule for a simple event: .9916 9936 9952 9964 Chapter 14- Multiple Regression Formulas for Chapter 14 along with the chapter are on the Web site for the text. 9990 9993 Chapter 15. Nonparametric Methods Formulas for Chapter 15 along with the chapter are on the Web site for the text. 9998 Empirical rule: For a specific bell-shaped distribution, about 68% of the ob servations fall in the interval (-a) to (+), about 95% fall in the interval (20) to (+20), and about 99.7% fall in the interval (x-30) to (+30) -Q,- First quartile given by the value of the middle term among the (ranked) observations that are less than the median Q-Second quartile given by the value of the middle term in a ranked data Page 10 of 13 Q-Third quartile given by the value of the middle term among the (ranked) observations that are greater than the median P₁-Value of the 100th term in a ranked data set - Percentile rank of a Number of values less than 100 Total number of values in the data set PE) Total number of outcomes Classical probability rule for a compound event PA) Number of outcomes in A Total number of outcomes - Relative frequency as an approximation of probability: P(A) = Condition for independence of events: P(A)=P(AB) and/or P(B)=P(BA) For complementary events: P(A) + PA) 1 Multiplication rule for dependent events: P(A and B)- P(A) P(BA) Multiplication rule for independent events: P(A and B)=P(A) P(B) . Conditional probability of an event P(A and B) P(A and B) P(A/B)= P(R) and P(4) P(A) Page 8 of 13
Transcribed Image Text:z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.5 1.6 1.7 1.8 20 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 5000 5398 5793 6179 3.2 3.3 6554 6915 7257 7580 7881 8159 8413 8643 8849 9032 9192 9332 9452 9554 9641 9713 9772 9821 9861 9893 9918 9938 9953 9965 9981 9987 00 -3.4 0003 0005 -3.2 0007 -3.1 0010 -3.0 -2.9 0019 -2.8 0026 -2.7 0035 0047 -2.5 0062 01 5040 5438 5832 6217 -14 0808 -1.3 0968 -1.2 1151 -1.1 1357 -1.0 1587 6591 6950 -0.9 1841 -0.8 2119 -0.7 2420 -0.6 2743 -0.5 7291 7611 7910 8186 8438 -0.4 3446 -0.3 3821 -0.2 4207 -0.1 0.01 5000 8665 8869 9049 9207 9345 9990 9993 9995 9997 This is Table IV of Appendix C. 9463 9564 9649 9719 9778 9826 9864 9896 9920 9940 9955 9966 9975 9982 9987 9991 9993 9995 9997 01 0003 0005 0007 0009 0013 -1.9 0287 0281 -1.8 0359 -1.7 0446 0018 0025 0034 0045 0060 -1.6 0548 0537 -1.5 0668 0655 0351 0436 0793 0951 1131 1335 1562 1814 2090 2389 2709 3050 3409 3783 102 4168 5080 4960 5478 5871 6255 6628 6985 7324 7642 7939 8212 8461 8686 8888 0082 0080 0078 0132 -2.3 0107 0104 0102 -2.2 0139 -21 -2.0 0228 0136 0174 0179 0170 0222 0217 9066 9222 9357 9474 9573 9656 9726 9783 9830 9868 9898 Table IV Standard Normal Distribution Table 9922 9941 9956 9967 9976 9982 9987 9991 9994 9995 9997 1002 0001 0005 0006 0009 0018 0024 0033 0044 0059 0274 0344 0427 0526 0643 0778 0934 1112 1314 1539 1788 2061 2358 2676 3372 3745 4522 4920 003 5120 5517 5910 6293 6664 7019 7357 7673 7967 8238 8485 8708 8907 9082 9236 9370 9484 9582 P(1) --- xC₂ Poisson probability formala P(x) 9834 9871 9901 9925 9943 9957 9968 9977 9495 9591 9664 9671 9732 9738 9788 9793 9983 9988 9991 9994 9996 9997 normal curve to the left of z with the values of z equal to 0 or positive. 03 0003 0004 0006 0009 0012 0017 0023 0032 0043 0057 0075 0099 0129 0166 0212 0268 0336 0418 0516 0630 0764 0918 1093 1292 1515 1762 2033 2327 3336 3707 4483 4880 Joint probability of two mutually exclusive events: PA and B) 0 . Addition rule for mutually nonexclusive events: P(AB)=P(A) + P(B)- PA and B) Addition rule for mutually exclusive events: P(AB)=P(A) + P(B) Chapter 5. Discrete Random Variables and Their Probability Distributions Mean of a discrete random variable x:=xP(x) . Standard deviation of a discrete random variable x: a-√2+¹P(1)-² 04 5160 5557 5948 6331 Chapter 6. Continuous Random Variables and the Normal Distribution 2 value for an x value Value of x when . o, and are known: x=+50 7054 Chapter 7. Sampling Distributions - Mean of : . Standard deviation of I when n/Ns .05 aya/V I-K 7389 7704 7995 8264 8508 8729 8925 9099 9251 factorial: w-1-2) 3-2-1 Number of combinations of a items selected x at a time: F.x-x) . Number of permutations of items selected x at a time: 04 0003 0004 0006 0008 0012 (-x) Binomial probability formulac P(x)-C, pq Mean and standard deviation of the binomial distribution: panday Hypergeometric probability formula: 9959 9969 9977 9984 0016 0023 0031 0041 0055 9992 9994 9996 9997 0073 0096 0125 xt . Mean, variance, and standard deviation of the Poisson prob ability distribution: =A²A, and VĀ 0207 0262 0329 0409 0505 0618 0749 0901 1075 1271 1492 2005 2296 2611 3300 3669 The entries in this table give the cumulative area under the standard normal curve to the left of z with the values of z equal to 0 or negative. 4443 9838 9842 9875 9878 9904 9906 9927 9945 4840 005 5199 5596 5987 6368 6736 7088 7422 7734 8023 8289 8531 9505 9599 9678 9744 9798 9929 9946 9960 9970 8749 8770 8944 8962 9115 9131 9265 9279 9394 9406 9984 9989 9992 9994 9996 9997 05 0003 0004 0006 0008 0011 0016 0022 0030 0040 0054 0071 0094 0122 0158 0202 0256 0322 0401 0495 0606 0735 0885 1056 1251 1469 1711 1977 2266 2578 2912 3264 3632 06 5239 4404 4801 56.36 6026 6406 6772 7123 7454 7764 8051 9515 9608 8315 8340 8554 8577 9686 9750 9803 9961 9971 9985 9992 9994 9996 9997 06 9846 9850 9881 9884 9909 9911 9931 9932 9945 9949 0003 0004 0006 000 0011 0015 0021 0039 0052 0091 0119 0197 0250 0314 0392 0485 0594 0721 0869 1038 1446 1685 1949 2236 2546 3228 3594 4364 1867 5279 3675 6064 6443 4761 7157 746 7794 8078 of 8980 9147 9292 9418 9525 9616 9693 9756 9808 9962 9972 9979 9985 9995 9996 9997 107 0003 0004 0005 0008 0011 0015 0021 0028 0038 0051 0089 0116 0150 0192 0244 0307 0384 0475 0582 0708 0853 1020 1423 3192 3557 108 08 5319 5714 6103 6480 7190 7517 7823 8106 8365 8599 - Margin of error of the estimate for a E20 of th - Determining sample size for estimating 8810 8997 9162 9306 9429 9535 9625 9854 9913 9934 9951 9963 9973 9956 0003 0004 0005 0007 9699 9706 9761 9767 9812 9993 9995 9996 0014 0020 where=1/√ñ 0037 0049 0066 0087 0113 0146 0188 0239 0301 0375 pt 2 where Margin of error of the estimate for p E-2 where - V Determining sample size for estimating p -²/E² 0465 0571 0694 0838 1003 1401 4681 5359 5753 6141 6517 6879 7549 8133 8389 8830 9015 Confidence interval for p for a large sample: V/n 9177 9319 9545 Chapter 9- Hypothesis Tests about the Mean and Proportion 9857 9890 9916 9936 9952 9964 9986 9990 9993 9995 9997 9998 09 0002 0003 0005 0007 0010 1660 1635 1922 1894 2206 2177 2148 2514 2483 2451 2843 2810 2776 0019 0048 0064 0084 3156 3520 3897 3859 4325 4286 4247 4721 4641 (continued on next page) 0110 0143 0183 0233 0294 0367 0455 0559 0681 0823 0985 1170 1379 - Population proportion: p-X/N Sample proportion: -x/ - Mean of p-p Standard deviation of when a/N05: Vpq/n 1611 1867 Chapter 8- Estimation of the Mean and Proportion - Point estimate of i .Confidence interval for using the normal distribution when is known 3121 3483 try where aya/V Confidence interval for a using the r distribution when or is Page 11 of 13 Test statistic z for a test of hypothesis about ja using the normal distribution when is know 2 where y Test statistic for a test of hypothesis about ja using the t dis tribution when or is not known -Test statistic for a test of hypothesis about p for a large sample: -- where - Page 9 of 13 M8 38333 88889 30212 9548A HAAaa aaaaa 3.1 3.3 00 3000 5398 3793 6179 6554 6915 7257 7580 8643 8849 9032 9192 9332 9452 9554 9641 9713 9772 7881 7910 81.59 8186 8413 8438 9861 9893 9918 9938 9953 9965 9974 01 9981 9987 3040 3438 5832 6217 6591 6950 7291 7611 3665 8869 9207 9345 9463 9564 9649 9719 9876 9864 9896 9920 9940 9955 9966 9975 9982 9987 9990 9993 9995 9995 9997 9997 This is Table IV of Appendix C. -- ww 9993 002 5080 Within-samples sum of squares 5478 5871 6255 6985 7324 7642 7939 8212 8461 Total sum of squares: 9726 9783 9830 Chapter 12- Analysis of Variance Let 9898 9922 9941 the mumber of different samples (orta) 9956 9967 9976 9982 9987 . Cumulative percentage 9991 9994 9995 8485 8686 8708 8888 81417 9066 9082 9222 9236 9357 9370 9474 9573 03 2²- .Confidence interval for the population variance o X- to Test statistic for a test of hypothesis about N Test statistic for a goodness-of-fit test and a test of inde- (0-1 5120 Zury- (Emyp 5517 6293 2004 the size of sample 7,- the sum of the values in sample i the number of values in all samples 7357 7673 7967 - ++ the sum of the values in all samples the sm of the squares of values in all samples For the F distribution and normal curve to the rest of z with the values of z equal to 0 or positive. 9484 9582 9664 9732 9788 Degrees of freedom for the somerator--1 Degrees of freedom for the denominator-k Between-samples sum of squares - Range - Largest value - Smallest value . Variance for ungrouped data (Ex)² IP- 9834 9871 9901 9925 SST SSB SSW-1 Variance between samples: MSB SSB/A-1) Variance within samples: MSW-SSW/(-A) Test statistic for a one-way ANOVA lest FMSB/MSW 9957 9968 9977 9983 99k 9991 9994 9996 9997 Chapter 13 Simple Linear Regression Simple linear regression model: y= A + B + e Estimated simple linear regression model: 9-a + bx Prem S. Mann Chapter 2- Organizing and Graphing Data . Relative frequency of a class f/2f N and ²- a² m - Standard deviation for grouped data: (Emy) Smy-N . Percentage of a class (Relative frequency) x 100 . Class midpoint or mark (Upper limit +Lower limit)/2 . Class width Upper boundary - Lower boundary . Cumulative relative frequency 04 - (Cumulative relative frequency) x 100 Chapter 3 - Numerical Descriptive Measures - Mean for ungrouped data: Ex/N and 7-x/m Mean for grouped data: Emf/N and I Σmf/n where is the midpoint and / is the frequency of a class Median for ungrouped data Value of the middle term in a ranked data set 2² (+)² 5100 5557 5948 6331 6700 7054 7389 7704 7995 8264 8508 8729 8925 Cumulative frequency Total observations in the data set 9251 9382 - 9495 9591 9671 9738 9793 and where is the population variance and is the sample Standard deviation for ungrouped data (Ex) 9838 9875 9904 9927 9945 In'y- (Imp)? 9959 9900 9992 9994 9996 9997 9984 9988 N and s #-1 where and s are the population and sample standard de viations, respectively Variance for grouped data Emp 05 5199 5596 3987 6368 6736 7088 -V Chebyshev's theorem: For any number & greater than 1, at least (1-1/2) of the values for any distribution lie within k standard deviations of the mean 7422 7734 8023 8289 8749 8944 9115 9265 9394 9505 9678 9744 9798 9842 9878 9906 9929 9946 06 3239 3636 6026 6772 7123 7454 7764 8051 8315 8554 8770 8962 9131 9279 9406 9515 9608 9686 9750 9803 9846 9881 9909 9931 9948 9961 9960 9970 9978 9984 9989 9989 9992 9994 9996 9997 9992 9994 9996 9997 Sum of squares of xy, x, and 55,-2-(X) 17 3279 3675 6064 6443 6808 7157 .Confidence interval for 92, where Prediction interval for 7486 7794 8078 8340 8577 KEY FORMULAS Introductory Statistics, Sixth Edition 8790 3980 9962 9972 9979 9979 9985 9985 9989 9292 9525 9616 9693 9756 9884 9911 9932 9949 9992 9995 9996 9997 3319 5714 6103 6430 08 6841 7190 7517 7823 8106 8365 8599 8810 www 9162 9306 9429 9533 960 9761 9812 9887 9913 9934 9951 Least squares estimates of A and B -55/55, and any-A Standard deviation of the sample en 9961 9973 9980 9986 9990 www. 59995 9996 9997 Error sum of squares SSE-1-2(y-99 (₂) Total sum of square SST-1²- Regression sun of squees SSRSST-SSE . Coefficient of determination: -55/55 .Confidence interval for B bt, where 44/V55, Test statistic for a test of hypothesis about B Linear correlation coefficient VSS, 55 Test statistic for a test of hypothesis about po (₂-19 99 5359 5753 6141 6517 Interquartile range: IQR-0-0 - The Ath percentile: 7224 7549 7852 8133 8621 9015 9177 9319 9545 9706 9767 9817 Chapter 4- Probability Classical probability rule for a simple event: .9916 9936 9952 9964 Chapter 14- Multiple Regression Formulas for Chapter 14 along with the chapter are on the Web site for the text. 9990 9993 Chapter 15. Nonparametric Methods Formulas for Chapter 15 along with the chapter are on the Web site for the text. 9998 Empirical rule: For a specific bell-shaped distribution, about 68% of the ob servations fall in the interval (-a) to (+), about 95% fall in the interval (20) to (+20), and about 99.7% fall in the interval (x-30) to (+30) -Q,- First quartile given by the value of the middle term among the (ranked) observations that are less than the median Q-Second quartile given by the value of the middle term in a ranked data Page 10 of 13 Q-Third quartile given by the value of the middle term among the (ranked) observations that are greater than the median P₁-Value of the 100th term in a ranked data set - Percentile rank of a Number of values less than 100 Total number of values in the data set PE) Total number of outcomes Classical probability rule for a compound event PA) Number of outcomes in A Total number of outcomes - Relative frequency as an approximation of probability: P(A) = Condition for independence of events: P(A)=P(AB) and/or P(B)=P(BA) For complementary events: P(A) + PA) 1 Multiplication rule for dependent events: P(A and B)- P(A) P(BA) Multiplication rule for independent events: P(A and B)=P(A) P(B) . Conditional probability of an event P(A and B) P(A and B) P(A/B)= P(R) and P(4) P(A) Page 8 of 13
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