1A) For which of the following scenarios would it be appropriate to use a one-sample z-test? - Comparing the number of packs of cigarettes smoked per day by a sample of residents of rural areas with the number of cigarettes smoked per day by all persons in the USA - Comparing the results of Likert scale patient satisfaction scores of patients at the VA against the Likert scale patient satisfaction scores of the overall population of patients - Comparing political affiliation scores (ranging from 0-20) of ASU students against the political affiliation scores of all college students - Comparing the average weight of newborns in Hospital A to the average weight of newborns in Hospital B 1B) Which of the following is not true about measures of effect? -You can compare measures of effects from different studies with the same outcome measure -They are affected by sample size -there are many different types of measures of effect -they aid in the interpretation of statistical results 1C) The purpose of the one-sample test is: (choose one or more) - Determine the probability that two groups are the same with respect to a particular variable -To examine the difference between one group and a population -To compare a test statistic to alpha -Compare two groups on a variable that is interval/ratio in measure 1C) Which of the following are assumptions of a one-sample test? (choose one or more) - ordinal or nominal scale of measurement - Interval or ratio scale of measurement - normal distribution of data - random sampling or selection 1D) How do we interpret Cohen's D of .61? - medium effect - small effect - No answer text provided. - large effect 1E) Which of the following is/are true about the standard error of the mean? (choose one or more) - It is a measure of variance - It is larger than the standard deviation of the population - It gets larger as sample size gets smaller - =population standard deviation divided by the square root of sample size
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In assessing the validity of any test of hypotheses, it is good practice to answer the following question: a) test the hypotheses at several different levels of significance b) examine the probability model by using exploratory data analysis on the data c) test both one- and two-sided hypotheses to help guarantee consistency d) All of the above
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A sample that does not provide a good representation of the population from which it was collected is referred to as a(n) biased sample.
Expand Your Knowledge: Logarithmic Transformations, Exponential Growth Model There are several extensions of linear regression that apply to exponential growth and power law models. Problems 22-25 will outline some of these extensions. First of all, recall that a variable grows linearly over time if it adds a fixed increment during each equal time period. Exponential growth occurs when a variable is multiplied by a fixed number during each time period. This means that exponential growth increases by a fixed multiple or percentage of the previous amount. College algebra can be used to show that if a variable grows exponentially, then its logarithm grows linearly. The exponential growth model is $y=\alpha \beta^{x}$, where $\alpha$ and $\beta$ are fixed constants to be estimated from data. How do we know when we are dealing with exponential growth, and how can we estimate $\alpha$ and $\beta$ ? Please read on. Populations of living things such as bacteria, locusts, fish, panda bears, and so on, tend to grow (or decline) exponentially. However, these populations can be restricted by outside limitations such as food, space, pollution, disease, hunting, and so on. Suppose we have data pairs $(x, y)$ for which there is reason to believe the scatter plot is not linear, but rather exponential, as described above. This means the increase in $y$ values begins rather slowly but then seems to explode. Note: For exponential growth models, we assume all $y>0$. $$ \begin{array}{|l|lrrrr} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline y & 3 & 12 & 22 & 51 & 145 \\ \hline \end{array} $$ Consider the following data, where $x=$ time in hours and $y=$ number of bacteria in a laboratory culture at the end of $x$ hours. (a) Look at the Excel graph of the scatter diagram of the $(x, y)$ data pairs. Do you think a straight line will be a good fit to these data? Do the $y$ values seem almost to explode as time goes on? (b) Now consider a transformation $y^{\prime}=\log y .$ We are using common logarithms of base 10 (however, natural logarithms of base $e$ would work just as well). Consider the following data, where $x=$ time in hours and $y=$ number of bacteria in a laboratory culture at the end of $x$ hours. (a) Look at the Excel graph of the scatter diagram of the $(x, y)$ data pairs. Do you think a straight line will be a good fit to these data? Do the $y$ values seem almost to explode as time goes on? (b) Now consider a transformation $y^{\prime}=\log y .$ We are using common logarithms of base 10 (however, natural logarithms of base $e$ would work just as well). $$ \begin{array}{l|lllll} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline y^{\prime}=\log y & 0.477 & 1.079 & 1.342 & 1.748 & 2.161 \\ \hline \end{array} $$ Look at the Excel graph of the scatter diagram of the $\left(x, y^{\prime}\right)$ data pairs and compare this diagram with the diagram in part (a). Which graph appears to better fit a straight line? (c) Use a calculator with regression keys to verify the linear regression equation for the $(x, y)$ data pairs, $\hat{y}=-50.3+32.3 x$, with sample correlation coefficient $r=0.882$. (d) Use a calculator with regression keys to verify the linear regression equation for the $\left(x, y^{\prime}\right)$ data pairs, $y^{\prime}=0.150+0.404 x$, with sample correlation coefficient $r=0.994$. The sample correlation coefficient $r=0.882$ for the $(x, y)$ pairs is not bad. But the sample correlation coefficient $r=0.994$ for the $\left(x, y^{\prime}\right)$ pairs is a lot better! (e) The exponential growth model is $y=\alpha \beta^{x}$. Let us use the results of part (d) to estimate $\alpha$ and $\beta$ for this strain of laboratory bacteria. The equation $y^{\prime}=a+b x$ is the same as $\log y=a+b x .$ If we raise both sides of this equation to the power 10 and use some college algebra, we get $y=10^{a}\left(10^{b}\right)^{x}$. Thus, $\alpha \approx 10^{a}$ and $\beta \approx 10^{b}$. Use these results to approximate $\alpha$ and $\beta$ and write the exponential growth equation for our strain of bacteria. Note: The TI-84Plus/TI-83Plus/TI-nspire calculators fully support the exponential growth model. Place the original $x$ data in list $\mathrm{L} 1$ and the corresponding $y$ data in list L2. Then press STAT, followed by $\mathbf{C A L C}$, and scroll down to option $\mathbf{0}$ : ExpReg. The output gives values for $\alpha, \beta$, and the sample correlation coefficient $r$.
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