i also ran a survey of students and determined that the college students prefer kikkoman to yamasa. green is not the middle of the hue color wheel. the point is that we are surrounded by data. 1 matrix of gray numbers on a white screen: print media variation 4 5 6 9 5 8 3 1 8 7 1 5 2 4 9 8 9 9 5 2 7 1 7 3 7 3 8 5 9 1 5 8 2 3 3 4 1 5 9 7 4 1 3 3 8 1 5 2 2 9 2 3 4 2 5 5 3 6 7 . a world of data and numbers. we could use the ratio of females to males in a class to estimate the ratio of females to males on campus. the sample size is the number of elements or measurements in a sample. inferential statistics: using descriptive statistics of a sample to predict the parameters or distribution of values for a population. the argument is a computer term for an input to the function. sampling refers to the ways in which random subgroups of a population can be selected. the sample is the experiments that are conducted. if a sample is a good random sample, representative of the population, then some sample statistics can be used to estimate population parameters. no mode is not the same as a mode of zero. if the mean was calculated from a sample then the mean is the sample mean. the median is the value that is the middle value in a sorted list of values. a line is drawn inside the box at the location of the 50th percentile. in some statistical programs potential outliers are marked with a circle colored in with the color of the box. as noted earlier, the range is one way to capture the spread of the data. the sample standard deviation is calculated in a way such that the sample standard deviation is slightly larger than the result of the formula for the population standard deviation. there is a greater probability of outliers in the population data. variables are said to be at the type and level of measurement of the data that the variable contains. remember that the mean is the result of adding all of the values in the data set and then dividing by the number of values in the data set. the z-scores of 4 and 1 would add to five. a distribution counts the number of elements of data in either a category or within a range of values. when the frequency data is calculated in this way, the distribution is not grouped into a smaller number of classes. the number of data values in each bucket is called the frequency. one of the aspects of a sample that is often similar to the population is the shape of the distribution. if there is no relationship, then there will be no shape to the pattern of the data on a graph. the column containing the x data is usually to the left of the column containing the y-values. the measurement that describes how closely to a line are the points is called the correlation. determining whether there is a relationship is best seen in an xy scattergraph of the data. the correlation r of − 0.93 is a strong negative correlation. in the example above the correlation r can be calculated and is found to be 0.06. zero would be random correlation. the total motion of the boat is in part due to the swell and in part due to the boy. and a is the y-intercept! this is the realm of the mathematics of probability. achromatopsia is controlled by a pair of genes, one from the mother and one from the father. a person with the combination aa is termed a carrier and has “normal” vision. what is the probability that a kosraen lives outside of kosrae? the complement of an event is the probability that the event will not occur. from the above raw data we can construct a two way table of results. the shape of the distibution of a sample is often reflective of the shape of the distribution of a population. there is also a way to recover the mean from the class values and the probabilities, although this depends on the class values being treated as being a part of a continuous distribution.

the result is an average age of 24.12 years for a resident of the fsm in 1994 and a standard deviation of 18.10 years. that the shape of the population distribution can be predicted by the shape of the distribution of a good random sample is important. the data for a section is gathered and tabulated. the area “under” each segment of the “curve” is the probability of a women being in that range of heights. 14.63% of the area is to the left of 60 inches. although this data is actually sample data and not population data, we will treat the mean and standard deviation as population parameters. note that the area is expressed as a decimal. examples include the population mean μ and the population standard deviation σ. a statistic is a numerical description of a sample. a relative frequency histogram of the sample means is plotted in a heavy black outline. the sample mean distribution is a heap shaped, as in the shape of the normal distribution, and centered on the population mean. note that the distribution of the sample means is narrower than the sample in the above example. the mean is composed of a sample of data values. the sample median can be a good point estimate for a population median, especially in situations where the data is not normally distributed. the narrower and taller line is the distribution of all possible sample means from that population. for the purposes of this course a 95% confidence interval is often used. the standard error of the mean is sx/√(n). the data seen in the table is the number of jumps for twenty-seven female jumpers. a capital p is used to refer to the population proportion. in the “real world” what often happens is that a result is found to not be statistically significant as the result of an initial study. in case ii a sample taken from the population is unlikely to produce the sample mean seen for that particular sample. in education and the social sciences there is a tradition of using a 95% confidence interval. the way to read that is to understand the μ as meaning “the sample could have a population mean of 25.4”. accepting the alternate hypothesis would be asserting that the population mean is the sample mean value. thus in hypothesis testing there is a tendency to utilize an alpha of 0.05 or 0.01 as a way to controlling the risk of committing a type ii error. the p-value is a calculation of the area “beyond” the t-statistic. in this text “surprising” is any p-value less than 0.05. if a result is surprising that means that the distance of the sample mean from the proposed population mean is surprisingly large, as in large enough to be statistically significant. that is a heck of a mathematical mess: the answer depends on your personal willingness to take a particular risk. because many studies in education and the social sciences are done at an alpha of 0.05, a p-value at or below 0.05 is used to reject the null hypothesis. the proper way to use a one-tailed test is to first do a two-tailed test for change in any direction. each element in the sample is considered as a pair of scores. the 95% confidence interval includes a possible population mean of zero. note too that while many paired t-tests for a difference of sample means involve pre and post data, there are situations in which the paired data is not pre and post in terms of time. the sample mean for the x1 data is x1. in case ii the difference in the sample means is too large for that difference to likely be zero. confidence intervals for each sample provide more information than a p-value and the declaration of a significant difference is more conservative. that is not being proved, a population mean difference of zero is taken as a given by the null hypothesis. even if there is a difference in the mean, that difference is not necessarily significant. the effect size can only be calculated if there is a significant difference in the means. if one is a researcher with some knowledge of statistics, then the questions to be asked will differ. what can be analyzed, what can be done, depends in part on how many variables are present and the level of measurement. a second possibility is that the data represents a “before-and-after” set of measurements. for both the paired data and independent samples data there is also the possibility that one could be testing for a difference of medians or a difference of variances (standard deviations). then there is a greater likelihood that looking for correlations among the columns might a useful approach.

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