Let the x-axis have the following intervals:

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1 1 & 2. For the following sets of data calculate the mean and standard deviation. Then graph the data as a frequency histogram on the corresponding set of axes. Set 1: Length of bass caught in Conesus Lake (cm) Length Frequency Let the x-axis have the following intervals: Mean: = (27.6) + 3(29.2) (30.4) = cm Standard Deviation: s = ( x x) i n ( xi x) = = ( ) 2 + 2( ) 2 + 3( ) 2 + ( ) 2 + n 1 2( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 = = = = s = 5.16 = Set 2: Mating age of twenty female lions, in months. Age (months)

2 Let the x-axis have the following intervals: Mean: = = Standard Deviation: = ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + 2( ) 2 + 2( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 = = = = s = = Data Set 3: Number of eggs laid by 20 hens in one week, just draw a graph and calculate the mean. Number of eggs Number of hens Let the x-axis go from 0-9 Mean: 6(1) + 4(2) + 3(3) + 2(4) = Do the values calculated for the standard deviation of bass lengths and lion mating ages correspond to degree of dispersion apparent in the graphs? ~ Yes. If we compare the two graphs, we see that the first one, which has a smaller standard deviation, has its points/values plotted very close to one another. While in the second graph, which has a much larger standard deviation, the values are spread out much more across the axis. 2. Look at each graph, which ones have symmetrically distributed data? Skewed?

3 ~ The first two have fairly symmetrically distributed data, however, the last graph, with the hens and their eggs, is fairly skewed. As you can see, the majority of the data is distributed towards the left side of the graph, and then rapidly, the values decrease. 3. Using your calculators, create a scatter plot of the following data: Linear Regression CO2 Level Year (ppm) CO2 Level (ppm) y = 1.505x Year Which type of regression would best fit this data? First, enter data into List 1 and 2 of your calculator. Then, go to STAT PLOT, and turn on plot 1. Chose to plot the scatter plot option, and enter L1 for the Xlist and L2 for the Ylist. Adjust the window such that the 1980 < x < 2010, and 340 < y < 380. Finally, graph. After examining the data, its clear to see that a linear regression will result with the best fit. Give the equation of this best fit curve: Exit the graph, and go into STAT and then to CALC. Choose option 4: LinReg (ax + b). Hit enter, and then type in L1, L2. Hit enter again. This should give you the appropriate a and b values to plug into the equation. Therefore, the best fit line equals Y = 1.505x Use this equation to estimate the CO2 level in 2008 Here, we just plug in 2008 for our x value of our best fit line: Y = (2008) = ppm What about in 1989?

4 Here, we do the exact same thing as above: Y = (1989) = ppm 4. Radio Active Decay: Polonium-210 has a half life of around 138 days, that is, in 138 days, half of the original sample will have decayed. Exponential Regression Sample Mass (g) y = e x Days Passed Mass (g) Days Graph the following data using your calculator and determine the best fit line, using an exponential curve. ~ Again, enter the data into L1 and L2 of your calculator under STAT, Edit. Then plot, graph, the points using stat plot, and adjust the window. A plot, similar to that above, should appear on your screen. After plotting the graph, go again to STAT, CALC and scroll down to 0:ExpReg. Hit enter, and then enter L1, L2, and hit enter again. Obtain the appropriate a and b values and place them into the equation for an exponential fit: y = * x Using this equation, what is the mass of the sample after 2 half lives? First we need to determine how many days are equivlanet to 2 half lives: 2 * 138 = 276 days Now we plug this into the equation: Y = * (0.9948) 276 = grams Given that the actual formula for determining the amount of decay is equal to:

5 m(t) = m 0 e -rt ln( 2) where r = ln2/h h = half life value (138) h Plugging in our known values results in: ln( 2) r = = m(276) = (500)e ( )(276) = grams Compare your result obtained from using this formula to that obtained by using the best fit curve. What can you conclude about your regression fit? The result obtained through interpolation of the data, using the best fit curve, is very close to that actual value obtained theoretically. As a result, one can conclude that the equation determined by the best fit curve is fairly accurate. 5 The National Cancer Society decided to take a poll in order to determine how much time the average American spends at the beach, sun bathing. The poll was conducted throughout the states of Florida, California, Virginia, and South Carolina. It was concluded that 60% of those polled spend at least 3 weeks of the year at the beach. - Can the National Cancer Society confidently conclude that based on this poll, that 60% of all Americans spend at least 3 weeks at the beach? Why or Why Not - What could you do to make this study more statistically valid? ~ No, the National Cancer Society can not confidently conclude that 60% of all Americans spend 3 weeks out of the year at the beach. Because the study was conducted only in coastal states, the results are based on a highly biased sample. Surely the people living in these states will spend more time at the ocean than those in the Midwest or northern states. Therefore, in order to make their study more statistically valid and accurate, the National Cancer Society should not only survey coastal states, but interior ones as well. Doing this would result in a much more random, non biased sample population.

6 6. A series of statistical test were conducted, and their results plotted. Based on the following graphs, which tests results showed positive strong or weak correlation? Negative strong or weak correlation? Zero Correlation? Weak Positive Correlation Zero Correlation Strong Negative Strong Positive Weak Negative Correlation Correlation Correlation

7 Brett Darrow 0020: Understand the principles, properties, and techniques of data analysis and statistics Measures of Central Tendency Central tendency refers to the value of a set of data that falls near the middle of the overall set. - measures the overall tendency, or trend of the data Three Measures: Mean also known as arithmetic mean, is equal to the sum of the numbers divided by n (where n = number of data values) Median When the data is arranged in numerical order, this value refers to that which falls in the very middle. Mode the value that appears most often in a set of data values.. Dispersion: o Measures the degree to which the data cluster around the middle o Measures include: Range the difference between the highest value and the lowest value of a set of data. Deviation how far a single value deviates from the mean value = (X i -X) Standard Deviation A measure of variability. Standard deviation measures the average distance of a data element from the mean. The following is the formula for sample standard deviation: 2 ( x x) i s = n 1 Variance A measure of variability given by the average of squared deviations. The following is the formula for sample variance 2 ( xi x) = n 1 o Normal Distribution Often just called the bell-curve or bell-shaped curve. A distribution of data that varies about the mean in such a way that the graph of its probability density function is a normal curve. The height of the curve is specified by the mean and standard deviation of the distribution. (See Attached for Picture) 68.2% of the values lie within one standard deviation of the mean 95.4% lie within two standard deviations 99.8% lie within three standard deviations of the mean Skewed Data: - Asymmetric distribution of the data values. The values on one side of the distribution curve tend to extend further from the "middle" than the values on the other side

8 Statistical Hypothesis and Testing: - Evaluating statistical claims made for a given set of data: - Sampling: o Population the complete collection of elements/values to be studied. o Sample a subset of a population o Random Sample (Simple Sample) occurs when the sampled values are selected independently from the same population. o Biased Sample a chosen sample of values that results in consistently overestimating or underestimating some characteristic of the population - Statistical Hypothesis: o Is an assertion or conjecture concerning one or more populations. o components of a formal hypothesis test: Null hypothesis (H 0 ) is a simple statement concerning the value or a predicted outcome of a population parameter (it is determined solely on the basis of whether or not is can be rejected). Ex: innocent till proven guilty is a null hypothesis Alternative hypothesis (H 1 ) is the statement that must be true if the null is false Conclusions: fail to reject the null or reject then null Statistical Test Interpretations: Such tests are conducted in order to determine whether or not the null hypothesis should be accepted or rejected, based on the resulting data. - Types of Tests: 1. t-test used to analyze data, measurements, which are continuously variable used to determine how statistically different the means of two sets of data are; distinguishes whether these differences occur naturally or are due to sampling error. If a large difference between the two is found, then there are two possible reasons why: 1. The two sample means are a poor reflection of the populations from which they were taken from. 2. Or the two sample means are in fact good reflections of their parent populations, and that our initial assumption, that the two populations are supposed to be closely related, is at fault. The larger the difference between the sample means, the less likely the first interpretations becomes, and the more we are inclined to believe the second.

9 Distribution Curve: Assumptions : - unknown population standard deviation value - sample size must be smaller than 30 (n < 0) - population to be sampled follows a normal distribution, roughly. 2. Chi-square test used to analyze non continuously measurable data (determine how many individuals fall into a certain category) - most widely used in inferential statistics, i.e. in statistical significance tests. It is useful because, under reasonable assumptions, easily calculated quantities can be proven to have distributions that approximate to the chi-square distribution if the null hypothesis is true. Distribution Curve Example: Assumptions : - The distribution is not symmetric - Values can be zero or positive, but not negative - As the number of degrees of freedom increases, the distribution approaches a normal distribution 3. Regression Analysis: Whenever possible, we try to approximate relationships between known quantities and tested quantities that are to be predicted in terms of mathematical equations. Whenever we use observed data to arrive at mathematical equations that describes the relationship between two variables, is known as curve fitting: Processes: - determine the appropriate type of curve that best fits the data - find the equation to this particular curve

10 - determine the accuracy and closeness of the fit, estimation approximations Least Squares Fit: - A procedure for finding the best-fitting curve to any given set of data points. - This is done by minimizing the sum of the squares of the residuals- the deviations from the fitted line to the observed values, of the points from the curve. - Linear Regression Equations: y = mx + b where: n( xy) ( x)( y) slope = m = n( x 2 ) ( x) 2 intercept = b = y m( x) n o Extrapolation: - the practice of constructing new data points, within the range, based on the determine regression model. o Interpolation: - the practice of predicting values, outside of the data range, based on the chosen regression model and equation. o Correlation: - Also called the correlation coefficient (r) - Indicates the strength and also direction of the relationship between two variables - The correlation coefficient always has a value between -1 and 1; with 1 indicating perfect positive correlation, -1 indicating a perfect negative correlation, and 0 indicating zero correlation. - Positive Correlation indicates a positive association between the two variables studied - Negative Correlation indicates a negative association. - Zero Correlation indicates no association exists between the variables. Stats on the TI (83): STAT function Can be used to make tables (L1, L2.) Various calculations can then be made from the entered data: Regresions: Linear, quadratic, cubic, exponential, power, logarithmic, etc Tests:

11 T-test, Z-test, chi-square LIST function: MATH: Min, max, mean, median, stddev, etc Note: When graph data with the TI 1. place data points into the lists, under the STAT, Edit option 2. STAT PLOT (2 nd Y=), turn Plot 1 on by hitting enter. Be sure to select the appropriate scatter plot type, and to enter the correct list for the Xlist and Ylist 3. Adjust axis, WINDOW 4. GRAPH, points should appear on screen 5. Determine the appropriate type of regression (see next page) and the appropriate a and b values. 6. Once you ve established your equation, enter it into Y1 (Y=) 7. Graph your best fit curve!

12 References: Webpages: (Basic Stat info) (LinReg Applet) (Regression Equations) /gloss.htm+statistics,+central+tendency,+skewed+data&hl=en&gl=us&ct=clnk&cd=3 (Stat Glossary) * \\Files\outbox\Math\West\Math Glossary\Glossary_D_Z_West doc (Dr. West s outbox, glossary) ATISTICS%20SITE (Another nice, understandable, general stat site) Books: Freund & Simon.(1997) Modern Elementary Statistics: Ninth Edition. New Jersey: Prentice Hal. Triola, (1998). Elementary Statistics: Seventh Edition: Addison-Weseley. Walpole, Myers, Myers, Ye. (2002) Probability and Statistics, for Engineers and Scientists Seventh Edition. New Jersey: Prentice Hall.

13 Linear Regression: y = ax + b - STAT - CALC - 4. LinReg enter - L1, L2 enter - Plug a and b values into equation Logarithmic: y = a + b ln x - STAT - CALC - 9: LnReg enter - L1, L2 enter - Plug a and b values into equation Exponential: y = Ce kx or y = ab x - STAT - CALC - 0:ExpReg enter - L1, L2 enter - Plug in a and b values Power: y = ax b - STAT - CALC - A:PwrReg.enter - L1, L2 enter - Plug in a and b

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