MATHAPALOOZA Finding Your Way around the Formula Sheet

MATH-A-PALOOZA! Finding Your Way around the Formula Sheet

STANDARD DEVIATION How spread out is the data? Is it consistent or sporadic?

STANDARD DEVIATION Which quarterback would you want on your fantasy football team?

STANDARD DEVIATION Calculate the mean of the data Make a data chart…step-by-step Subtract and square it…list it Count up sample size (-1)

STANDARD DEVIATION Calculate the mean of the data Make a data chart…step-by-step Subtract and square it…list it Count up sample size (-1) xi xi – xmean (xi - xmean)2 Length of Beetles (cm) 2. 0 -. 24 . 0576 2. 4 . 16 . 0256 2. 1 -. 14 . 0196 2. 5 . 26 . 0676 2. 2 -. 04 . 0016 ------------- Sum = __. 172 Sum = 11. 2 n= 5

S = √. 172 4 S =. 207 Fairly low variance, our data set is closely spaced

CHI-SQUARE What is this even for? ? ? To determine if your data is in fact close enough to the expected result that you can just say they are the same What is a NULL hypothesis? � Statement that the observed and expected should be the same (IV does not affect DV) If chi-square # is LARGER than the “cut-off”, then there is a LARGE difference between observed & expected. Should you ACCEPT or REJECT your null hypothesis?

CHI-SQUARE Roll the dice twice! 21 possible rolls…(11 possible values) Null hypothesis is…? � Each combination should be equally likely (the dice are not fixed!) Chi-square = 9. 63 What is the degrees of freedom? Which probability to use? What is critical value? (the “cut-off”) Is chi-square value higher or lower?

CHI-SQUARE A cellular biologist wants to verify that cells spend 90% of their time in interphase and only 10% in mitosis. Her null hypothesis states that 90% of the observed cells should be in interphase, while only 10% should be in mitosis. She prepares cells from onion root tips. Using a microscope, she counted 1000 cells from her preserved specimen. Her data are shown below. Calculate the X 2 to the nearest hundredth. Stage of the Cell Cycle Interphase Number of Cells Observed 872 Mitosis 128 Number of Cells Expected

CHI-SQUARE p 0. 05 0. 01 1 3. 84 6. 64 Degrees of Freedom 2 5. 99 9. 32 3 7. 82 11. 34 4 9. 49 13. 28 5 11. 07 15. 09 What are your expected values? Why? Calculate (O-E)… and then go ahead and square it…record each Divide that by E… record each Take the sum of last column…this is your chi-square value What is the df? What is the p? What is your critical value (cut-off)? Should you accept or reject the null hypothesis? Why? Stage of the Cell Cycle Interphase Number of Cells Observed 872 Number of Cells Expected 900 Mitosis 128 100 6 12. 59 16. 81 7 14. 07 18. 48 8 15. 51 20. 09

HARDY-WEINBERG Only works for a non-evolving populations Frequencies of alleles & genotypes in gene pool remain constant Must meet these requirements: �Very large population size �No migration �No net mutations �Random mating �No natural selection

HARDY-WEINBERG “p” is frequency of one allele; “q” is other p+q=1 When gametes combine, 2 alleles join… These are the organisms: p 2 + 2 pq + q 2 = 1 ALWAYS make a key (p, q, p 2, 2 pq, q 2) ALWAYS start by finding q or q 2 …why? ALWAYS re-check the question!

HARDY-WEINBERG In a population in H-W equilibrium, 36% of the individuals show the recessive trait. What is the frequency of the dominant allele? Where to start? What is “given”? �q 2 =. 36 �So q = 0. 6 �So 1 -q=p …p=0. 4 �Frequency of dominant allele = 0. 4

HARDY-WEINBERG In a population with two alleles for a particular locus, B and b, the allele frequency of B is 0. 7. What would be the frequency of heterozygotes if the population were in H-W equilibrium? �What is “given”? �p = 0. 7 �so, q = 0. 3 �Heterozygotes = 2 pq � 2 pq = 2(0. 7)(0. 3) = 0. 42

RATE & SLOPE To measure rate/slope… Y 2 – Y 1 X 2 – X 1 …just pick two points on the line if not specified What is the rate of growth in the pea plant between weeks 1 & 3? 60 – 10 3 – 1 = 25 cm/wk

RATE & SLOPE What is the cost per production? 90, 000 -60, 000 25, 000 -15, 000… = $3 per product

TEMPERATURE COEFFICIENT – Q 10 Why is regulating temperature so important? Denaturing proteins – too HOT = proteins unfold Metabolism rates (Q 10) - too COLD = reactions too slow

TEMPERATURE COEFFICIENT Many ectothermic organisms have a Q 10 value of 2. 0, but that may not always be the case. Always follow the information given. When given a choice…always choose two values that are 10 degrees apart… why? Use the higher temperature values as your “k 2” and “t 2” …why?

TEMPERATURE COEFFICIENT If the metabolism rate of a certain ectothermic organism is 28 units at 25 o. C, what will the rate be at 15 o. C? (Many ectothermic organisms have a Q 10 value of 2. 0) What is the exponent value? � 1 …because 25 -15 = 10 …and 10/10 = 1 How to find k 1? �Has to be half the value of k 2, if Q 10 = 2. 0 �So, k 1 = 14 units of metabolism

TEMPERATURE COEFFICIENT What is the Q 10 value for an organism that has a metabolic rate with a value of 39 units at 27 o. C and a metabolic rate of 41 units at 37 o. C? What is the exponent value? � 1 …because 37 -27 = 10 …and 10/10 = 1 How to find Q 10? �Divide … 41/39 = 1. 05

BIOMASS Total dry mass (take out the water and measure the mass) Must get uniform values (exponents may NOT match) Write out the value of each number without the exponents Look for the total biomass – don’t care about other numbers …this is our total “budget”.

BIOMASS What percent of biomass in this community is within the grasses? Look for total biomass – don’t care about other numbers above it! What is that total? 1. 8 x 10, 000 = 18, 000

BIOMASS Total Biomass = 18, 000 Subtract each listed value from total biomass. Whatever is leftover is the total for the grasses. � 18, 000 – 12, 000 – 4400 = 1600 But, why is this NOT the answer? Calculate percentage: 1600/18000 = 0. 0888888 � How to record? � = 9% or 8. 9% (however they ask!)

GIBBS FREE ENERGY How much energy is available Important to determine if chemical reactions will take place ΔG = ΔH – TΔS G = Free Energy H = Enthalpy T = Temperature in Kelvin S = Entropy Don’t forget to change the temperature to Kelvin! (add 273 to Celsius value…is listed on formula sheet!)

GIBBS FREE ENERGY An experiment determined that when a protein unfolds to its denatured (D) state from the original folded (F) state, the change in Enthalpy is ΔH = H(D) – H(F) = 46, 000 joules/mol. Also the change in Entropy is ΔS = S(D) – S(F) = 178 joules/mol. At a temperature of 20⁰C, calculate the change in Free Energy ΔG, in j/mol, when the protein unfolds from its folded state. ΔG = (46, 000) – (293) (178) ΔG = 46, 000 – 52, 154 = 6154 joules/mol

WATER POTENTIAL Water wants to move from high to low osmotic pressure Water potential = pressure + solute…if open to the air, pressure = 0 Solute potential will be: zero or LOWER (negative) Pressure potential will be zero or HIGHER (positive) the “i” will be 1 for sugars, 2 for salt ALWAYS draw the cells!

WATER POTENTIAL In plants, this typicaly means, water moves from soil roots shoots air (transpiration!) Be careful with the negatives! (always moves from high to low!)

ALWAYS DRAW the CELLS!!! If water potential is greater inside the cell than outside, then water will exit the cell Plasmolysis results (plasma membrane separates from cell wall)

ALWAYS DRAW the CELLS!!! If water potential is lower inside the cell than outside, then water will enter the cell Cell becomes turgid

WATER POTENTIAL What is the water potential of a cell with a solute potential of -0. 67 MPa and a pressure potential of 0. 43 MPa? � -0. 67 + 0. 43 = -. 24 MPa Cell A has a solute potential of -2. 0 MPA and a pressure potential of 0. 5 MPA. Cell B has a solute potential of -4. 0 MPa and a pressure potential of 0. 9 MPA. Which way will water flow when the two cells are placed against each other? �Cell A -2. 0 + 0. 5 = -1. 5 �Cell B -4. 0 + 0. 9 = -3. 1 �Which is more negative? Water moves from cell A to cell B

WATER POTENTIAL Calculate the water potential of a 0. 15 M sucrose solution. Assume a temperature of 27 C. Note: Solute potential = - i. CRT Pressure potential in open air is 0. i = 1 for sugars! add 273 to Celsius number! = - (1) (0. 15) (0. 0831) (300) = - 3. 74

WATER POTENTIAL A wind borne pollen grain with solute potential of -3. 0 MPa dried out somewhat after being blown about; this caused its turgor pressure (pressure potential) to go to 0. It lands on a flower stigma whose cells have a solute potential of -3. 0 MPa and a pressure potential of 1 MPa. Which way will water flow? From the pollen grain to the stigma, or the stigma to the pollen grain? Pollen -3. 0 + 0 = -3. 0 Stigma -3. 0 + 1 = -2. 0 Stigma is more negative…water leaves stigma & enters pollen

POPULATION GROWTH How much is this population growing? Or declining? Rarely have ACTUAL birth (B) and death (D) numbers, but this is what is provided on your formula chart! Growth Rate = d. N/dt = B – D

POPULATION GROWTH The growth rate may be calculated (or provided)… labeled as “r” Populations that are not limited in growth are growing exponentially Growth = r. N But, of course, eventually that carrying capacity (K) is reached because there always limits! Growth = r. N x [(K-N)/K]

POPULATION GROWTH A population of rats produces 224 new babies in a year. In the same time period, 124 of the rats died. What is the growth rate of this population? Growth = B – D = 224 – 124 = 100 rats/year A population of 1000 roaches invades a new habitat with virtually unlimited resources. They have a growth rate of 6. 12. What will their population total be after one year? d. N/dt = r. N = 6. 12 x 1000 = 6120

POPULATION GROWTH If the carrying capacity is 2400 and r=2. 1, what is the growth rate for a population of 1000 individuals? Growth = (2. 1)(1000) x [(2400 – 1000)/2400] = 2100 x 1400/2400 = 2100 x 0. 5833 = 1224

STANDARD ERROR OF THE MEAN A measure of how likely your mean value is to represent the overall set of data points Numerical values likely to be provided…just plug in! Which set of data has more variability?

STANDARD ERROR OF THE MEAN Ex: What is the SEM for James (from the fantasy football example)? SEM = 41. 1 / 4 = 10. 275

SURFACE AREA & VOLUME In a diffusion lab, agar cubes are used to illustrate the relationship of surface area/volume/diffusion rate. Fill in the values missing in the table. Block cm x cm A 2 x 2 B 3 x 3 C 4 x 4 Surface Area cm Volume ml Surface Area: Volume Ratio

SURFACE AREA & VOLUME If you put each of the blocks into a solution, into which block would that solution diffuse throughout the entire block fastest? Slowest? How do you explain the difference? Describe the relationship between the surface area: volume ratio and the percentage of the cube not affected by diffusion. Block cm x cm Surface Area cm Volume ml Surface Area: Volume Ratio A 2 x 2 24 8 3: 1 B 3 x 3 54 27 2: 1 C 4 x 4 96 64 1. 5: 1

OTHER MATH - PYRAMIDS How much energy is in each level of this ecosystem? Use 10% Rule unless otherwise specified! Simply move the decimal place one place to the left for every level you move up

OTHER MATH – MARK & RECAPTURE This is used to estimate population sizes of animals where individuals are highly mobile. It is of no value where animals do not move or move very little. The number of animals caught in each sample must be large enough to be valid. First capture each animal is marked in a distinctive way � Then the marked animals are released into the natural habitat and left for a period of time. Second capture only a portion of the second capture sample will have marked animals

OTHER MATH – MARK & RECAPTURE # in first sample (all marked) x total # in 2 nd sample Total population = # of marked in the second sample (recaptured) 30 turtles are captured in 1 km 2, they are marked and released back into the wild. Two weeks later 30 more turtles are caught. 6 had the marking of the original population. Based on this information, what is the best estimation of the turtle population in the area? 30 x 30 = 900/6 = 150

OTHER MATH – CROSSOVER FREQUENCY In fruit flies, long wings (A) and gray bodies (B) are dominant to vestigial wings and black bodies. In a cross of Aa. Bb x aabb. These are your results- 123 long wing, gray body 21 long wing, black body 27 vestigial wing, gray body 129 vestigial wing, black body Calculate the cross over value (recombination frequency) for the offspring of the test cross.

OTHER MATH – CROSSOVER FREQUENCY 123 long wing, gray body 21 long wing, black body 27 vestigial wing, gray body 129 vestigial wing, black body You recognize that this is recombination because you don’t get the expected 1: 1: 1: 1 ratio. To calculate Rf = number of recombinants/Total = 48/300 = 16% Rf This also tells you that the gene for wing length and color are 16 map units apart Could you complete a chi-square for this? !?

OTHER MATH – CROSSOVER FREQUENCY There are 4 genes on a single chromosome: A, B, C and D. They exhibit the following crossing over frequencies: A-B = 35% B-C = 10% C-D = 15% C-A =25% D-B=25% Determine the order of the genes on the chromosome. Start with the genes the farthest apart, and everything else usually falls into place. ADCB or BCDA (either is acceptable)

COMBINE H-W WITH CHI-SQUARE! In a certain species of flowering plant, the red allele R is dominant to the white allele, r. A student carried out a cross between a red flowered plant and a white flowered plant. When planted, the 158 seeds that were produced from the cross matured into 92 plants with red flowers and 66 plants with white flowers. Calculate the chi squared value for the null hypothesis that the red flowered parent was a hybrid for the flower color gene. Give your answer to the nearest tenth.

COMBINE H-W WITH CHI-SQUARE! In this cross Rr x rr you would expect a 1: 1 ratio Observed 92 red 66 white Expected 79 red 79 white O-E 13 -13 O-E 2 169 O-E 2/ E 2. 14 Chi squared = 4. 3 What is the df? The p? The critical value? What was the null hypothesis? Should you accept or reject it? Why?

THE END Resources used include: Campbell Biology Student Study Guide Campbell Biology - diagrams & pictures AP College Board – Biology Formula Sheet AP College Board – released questions Har-Ber HS - Math Review document