invest_3ed.pdf

# D describe what the central limit theorem says about

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(d) Describe what the Central Limit Theorem says about the distribution of sampling proportions in this context, assuming the null hypothesis is true, and produce a well-labeled sketch to illustrate your description. (e) In your graph, shade the area under the curve corresponding to the sample proportion of “hits” being 0.322 or higher.

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Chance/Rossman, 2015 ISCAM III Investigation 1.8 72 The area you have shaded represents the probability of a sample proportion exceeding 0.322, under the assumption that the subjects have no psychic ability (with n = 329 sessions). (f) Based on your shading, provide a rough guess of this area as a percentage of the total area under this normal curve. (g) How many standard deviations (SDs) above the mean is the value 0.322, the observed sample proportion of hits? [ Hint : First subtract the mean from 0.322, then divide the result by the SD.] Definition: The standardized score of an observation determines the number of standard deviations between the observation and the mean of the distribution: z = observation mean = x ± P standard deviation V This quantity is also referred to as the z -score. By converting to this z -score, we say we have standardized the observation. This provides us another metric or ruler for how unusual an observation is. The sign of the z -score tells us whether the observation falls above or below the mean. If X follows a normal distribution with mean P and standard deviation V , then Z follows a normal distribution with mean 0 and standard deviation 1. Because this z -score is larger than 2, we already know the probability to the right is rather small. But how can we determine this probability more precisely? Generally we would integrate the function over the region of interval (0 .322, ∞). Because that is not possible with the normal probability function, we will instead use technology, which implements numerical integration techniques to calculate this area/probability. Technology Detour Calculating Normal Probabilities In R: The iscamnormprob function takes the following inputs: x xval = the x value of interest x mean = the mean of the normal distribution x sd = the standard deviation of the normal distribution x direction = “above,” “below,” “outside” or “between” (using the quotes) x label = a string of text (in quotes) to put on the horiziontal axis x xval2 = an optional input for use with “outside” and “between” directions. x digits = specifies how many significant digits to display, default is 4 For example (with or without input labels): iscamnormprob(xval=.322, mean=.25, sd=.02387, direction="above", label="sample proportions") In Minitab x Choose Graph > Probability Distribution Plot (View Probability) x Keep the Distribution pull-down menu set to Normal and specify the values for the mean and standard deviation.
Chance/Rossman, 2015 ISCAM III Investigation 1.8 73 x Press the Shaded Area tab. Define the Shaded Area by X Value, specify the tail direction, and enter the observation value of interest. Press OK .

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