# Statistics Cheat Sheet: Key Concepts and Formulas

Find the probability that exactly four of the sampled people own a tablet computer. We input 15 for *n*, .3 for *p*, and 4 for *x* = **binompdf(15,.3,4) **15 is sample size .30=30% || To compute P(*x*), select **poissonpdf** and enter the values for λ*t* and *x* separated by commas and press ENTER. || To compute P(Less than or equal to *x*), select **poissoncdf** and enter the values for λ*t* and *x* separated by commas and press ENTER. || Let A and B be events with P(A) = 0.7, P(B) = 0.4, and P(B|A) = 0.2. Find P(A and B). = .14 || A fair coin is tossed four times. What is the probability that the sequence of tosses is HHHT? = .0625 || If P(A) = 0.21, P(B) = 0.81, and P(A and B) = 0.45, find P(A or B). = .57 || A student takes a true-false test that has 14 questions and guesses randomly at each answer. Let X be the number of questions answered correctly. Find P(Fewer than 4) = .0287 || The Australian sheep dog is a breed renowned for its intelligence and work ethic. It is estimated that 35% of adult Australian sheep dogs weigh 65 pounds or more. A sample of 11 adult dogs is studied. What is the mean number of dogs who weigh 65 lb or more? = 3.85 || **Normal distributions** have one mode. **Normal distributions** are symmetric around the mode. || **The mean and median of a normal distribution** are both equal to the mode. In other words, the mean, median, and mode of a normal distribution are all the same. The normal distribution follows the Empirical Rule || we found the area between *z* = −1.45 and *z* = 0.42. Find this area by using technology. **normalcdf **command. We enter the left endpoint of the interval (−1.45), the right endpoint (0.42), the mean (0), and the standard deviation (1) || we found the **area to the right of ****z**** = −0.58 normalcdf** command. We enter the left endpoint of the interval (−0.58). Since there is no right endpoint, we enter 1E99. Then we enter the mean (0) and the standard deviation (1) **NORMALCDF(-.58,1E99,0,1)** || **we found the ****z****-score** that has an area of 0.26 to its left. The *z*-score is found by using the **invNorm **command. We enter the area to the left (.26), the mean (0), and the standard deviation (1) **INVNORM(.26,0,1) **|| *z*-score that has an area of 0.68 to its right = 1-.68= **INVNORM(.32,0,1) **|| Let *x* be a value from a normal distribution with mean μ and standard deviation σ. The **z****-score** of *x* is *z*= (*x*-μ)/σ || IQ scores have a mean of 100 and a standard deviation of 15. Use technology to find the 90th percentile of IQ scores; in other words, find the IQ score that separates the upper 10% from the lower 90%. **INVNORM(.90,100,15) **|| finding the area to the right of *x* = 280 with μ = 272 and σ = 9 = **NORMALCDF(280,1E99,272,9) ||** finding the normal value that has an area of 0.98 to its left, where μ = 100 and σ = 15 **INVNORM(.98,100,15) || **

A sample of size 50 will be drawn from a population with mean 10 and standard deviation 8.

Find the probability that x will be between 8 and 11 = normalcdf(8,11,10, 8/√(50))

**Correlation coeffecient** = Linreg on calc then r=coefficient. || When we are given the value of the explanatory variable, we can use the least-squares regression line to predict the value of the outcome variable. || The least-squares regression line passes through the point of averages || The least-squares regression line DOES NOT predict the result of changing the value of the explanatory variable. || **Least Squares Regression Line ** ŷ = b0 + b1x**. **b1=slope b0=y intercept || Do not use the **least-squares regression** line to make predictions for x-values that are outside the range of the data. The linear relationship may not hold there. || Given a point (x, y) on a scatterplot, and the least-squares regression line ŷ = b0 + b1x, the residual for the point (x, y) is the difference between the observed value y and the predicted value ŷ. Residual = y − ŷ || When a residual plot exhibits no noticeable pattern, the least-squares line may be used to describe the relationship between the variables. || When a scatterplot contains outliers: 1) Compute the least-squares regression line both with and without each outlier to determine which outliers are influential. 2)Report the equations of the least-squares regression line both with and without each influential point. || The **coefficient of determination** is r², the square of the correlation coefficient r. || The coefficient of determination r² measures the proportion of the variation in the outcome variable that is explained by the least-squares regression line. The larger the value of r², the closer the predictions made by the least-squares regression line are to the actual values, on average. To **compute the coefficient of determination**, first compute the correlation coefficient r, then square it to obtain r². || The closer r² is to 0, the closer the predictions made by the least-squares regression line are to the actual values, on average= **FALSE** || **Bivariate data** are data that consist of ordered pairs. A scatterplot provides a good graphical summary for bivariate data. When large values of one variable are associated with large values of the other, the variables are said to have a positive association. When large values of one variable are associated with small values of the other, the variables are said to have a negative association. When the points on a scatterplot tend to cluster around a straight line, the relationship is said to be linear. || The **correlation coefficient** r measures the strength of a linear relationship. The value of r is always between −1 and 1. || A **plot of residuals** versus values of the explanatory variable is called a **residual plot**. When a residual plot has no apparent pattern, a linear model is appropriate. The **correlation coefficient** can be misleading in this regard, because the correlation may be large even when the relationship is not linear. || The **least-squares regression** line should be used for predictions only for values of x that lie within the range of the data used to compute the equation of the least-squares line. Making predictions outside the range of the data is called **extrapolation**, and such predictions are generally unreliable. || The **probability** of an event is the proportion of times the event occurs in the long run, as a probability experiment is repeated over and over again. || A **sample space** contains all the possible outcomes of a probability experiment. || An **event** is an outcome or a collection of outcomes from a sample space. || An **unusual event** is one whose probability is small= P<.05 the=””>**Empirical Method** consists of repeating an experiment a large number of times, and using the proportion of times an outcome occurs to approximate the probability of the outcome. || **General addition rule **P(A or B) = P(A) + P(B) − P(A and B) || Two events are said to be **mutually exclusive** if it is impossible for both events to occur. || **Addition rule for mutually exclusive events **P(A or B) = P(A) + P(B) || If A is any event, the **complement** of A is the event that A does not occur. The **complement** of A is denoted Ac. || Two events are **independent** if the occurrence of one does not affect the probability that the other event occurs. || **General Multiplication Rule **P(A and B) = P(A)P(B | A) or P(A and B) = P(B)P(A | B) || **Multiplication Rule for Independent Events** P(A and B) = P(A)P(B) || A **random variable** is a numerical outcome of a probability experiment. || **Discrete random variables** are random variables whose possible values can be listed. The list may be infinite — for example, the list of all whole numbers. || **Continuous random variables** are random variables that can take on any value in an interval. The possible values of a continuous variable are not restricted to any list. || **Computing the mean and standard deviation of a discrete random variable **ti-84 input L1 and L2 and do 1-VAR stats || **Binomial Distribution** – fixed number of trials are conducted, only two possible outcomes – Success or Failure ||