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load the data (file: session4data.rdata) make a new summary data frame (per subject and time) containing: the number of trials the number correct trials (absolute and relative) the mean TTime and the standard deviation of TTime the respective standard error of the mean keep the information about Sex and Age_PRETEST make a plot with time on the x-axis and TTime on the y-axis showing the means and the 95\% confidence intervals (geom_pointrange()) add the number of trials and the percentage of correct ones using geom_text()

load the data (file: session4data.rdata) make a new summary data frame (per subject and time) containing: ]]>%]]> %]]> ]]>

Situation H_0 is true H_0 is false ConclusionH_0 is not rejected Correct decision Type II error H_0 is rejected Type I error Correct decision

n number of observations (sample size)K number of samples (each having n elements)alpha level of significance nu degrees of freedommu population meanxbar sample meansigma standard deviation (population)s standard deviation (sample)rho population correlation coefficientr sample correlation coefficientZ standard normal deviate

attachment:twosided.pdf attachment:greater.pdf attachment:less.pdf The p-value is the probability of the sample estimate (of the respective estimator) under the null. The p-value is NOT the probability that the null is true.

The z-test is a something like a t-test (it is like you would know almost everything about the perfect conditions. It uses the normal distribution as distribution of the test statistic and is therefore a good example. To investigate the significance of the difference between an assumed population mean $\mu_0$ and a sample mean $\bar{x}$It is necessary that the population variance $\sigma^2$ is known. The test is accurate if the population values are normally distributed. If the population values are not normal, the test will still give an approximate guide.

Write a function which takes a vector, the population standard deviation and the population mean as arguments and which gives the Z score as result. name the function ztest or my.z.test - not z.test because z.test is already used set a default value for the population mean add a line to your function that allows you to process numeric vectors containing missing values! the function pnorm(Z) gives the probability of $$x \leq Z$$ Change your function so that it includes the p-value (for a two sided test) as result. now let the result be a named vector containing the estimated difference, Z, p and the n. You can always test your function using simulated values: rnorm(100,mean=0) gives you a vector containing 100 normal distributed values with mean 0.

Write a function which takes a vector, the population standard deviation and the population mean as arguments and which gives the Z score as result. ]]> ]]> ]]> Add a line to your function that allows you to also process numeric vectors containing missing values! ]]> The function pnorm(Z) gives the probability of $$x \leq Z$$ . Change your function so that it has the p-value (for a two sided test) as result. ]]> ]]> ]]> Now let the result be a named vector containing the estimated difference, Z, p and the n. ]]> ]]> ]]>

Z-test for two population means (variances known and equal) Z-test for two population means (variances known and unequal) To investigate the statistical significance of the difference between an assumed population mean $\mu_0$ and a sample mean $\bar{x}$. There is a function z.test() in the BSDA packageIt is necessary that the population variance $\sigma^2$ is known. The test is accurate if the population is normally distributed. If the population is not normal, the test will still give an approximate guide.

Now sample 100 values from a Normal distribution with mean 10 and standard deviation 2 and use a z-test to compare it against the population mean 10. What is the p-value? Now do the sampling and the testing 1000 times, what would be the number of statistically significant results? Use replicate() (which is a wrapper of tapply()) or a for() loop! Record at least the p-values and the estimated differences! Use table() to count the p-vals below 0.05. What type of error do you associate with it? What is the smallest absolute difference with a p-value below 0.05? Repeat the simulation above, change the sample size to 1000 in each of the 1000 samples! How many p-values below 0.05? What is now the smallest absolute difference with a p-value below 0.05?

Now sample 100 values from a Normal distribution with mean 10 and standard deviation 2 and use a z-test to compare it against the population mean 10. What is the p-value? What the estimated difference? ]]> ]]> ]]> Now do the sampling and the testing 1000 times, what would be the number of statistically significant results? Use replicate() (which is a wrapper of tapply()) or a for() loop. Record at least the p-values and the estimated differences! Transform the result into a data frame.

using replicate() ]]> ]]> ]]> using replicate() II ]]> ]]> ]]> using for() ]]> ]]> ]]> ]]> ]]> Use table() to count the p-vals below 0.05. What type of error do you associate with it? What is the smallest absolute difference with a p-value below 0.05? ]]> ]]> ]]> Repeat the simulation above, change the sample size to 1000 in each of the 1000 samples! How many p-values below 0.05? What is now the smallest absolute difference with a p-value below 0.05? ]]> ]]> ]]> Repeat the simulation above, change the sample size to 1000 in each of the 1000 samples! How many p-values below 0.05? What is now the smallest absolute difference with a p-value below 0.05? ]]> ]]>

Concatenate the both resulting data frames from above using rbind() Plot the distributions of the pvals and the difference per sample size. Use ggplot2 with an appropriate geom (density/histogram) What is the message?

Concatenate the both resulting data frames from above using rbind() Plot the distributions of the pvals and the difference per sample size. Use ggplot2 with an appropriate geom (density/histogram) ]]> ]]> ]]> ]]> attachment:hist.png Plot the distributions of the pvals and the difference per sample size. Use ggplot2 with an appropriate geom (density/histogram) ]]> ]]>

attachment:dens.png attachment:point.png attachment:dens2d.png

A t-test is any statistical hypothesis test in which the test statistic follows a Student's t distribution if the null hypothesis is supported. one sample t-test: test a sample mean against a population mean $$t = \frac{\bar{x}-\mu_0}{s/\sqrt{n}}$$ where $\bar{x}$ is the sample mean, s is the sample standard deviation and n is the sample size. The degrees of freedom used in this test is n-1

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There are two ways to perform a two sample t-test in R: given two vectors x and y containing the measurement values from the respective groups t.test(x,y) given one vector x containing all the measurement values and one vector g containing the group membership t.test(x $\sim$ g) (read: x dependend on g)

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if not stated otherwise t.test() will not assume that the variances in the both groups are equal if one knows that both populations have the same variance set the var.equal argument to TRUE to perform a student's t-test

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the t-test, especially the Welch test is appropriate whenever the values are normally distributed it is also recommended for group sizes > 30 (robust against deviation from normality)

use a t-test to compare TTime according to Stim.Type, visualize it. What is the problem? now do the same for Subject 1 on pre and post test (use filter() or indexing to get the resp. subsets) use the following code to do the test on every subset Subject and testid, try to figure what is happening in each step:\tiny make plots to visualize the results. how many tests have a statistically significant result? How many did you expect? Is there a tendency? What could be the next step?

use a t-test to compare TTime according to Stim.Type, visualize it. What is the problem? ]]> ]]> now do the same for Subject 1 on pre and post test (use filter() or indexing to get the resp. subsets) ]]> ]]> make plots to visualize the results ]]> ]]> how many tests have an statistically significant result? How many did you expect? ]]> ]]> ]]>

What could be the next step? attachment:excerchypo.pdf