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Hypothesis Test Conclusion Our chosen risk of a Type I error is called the criterion or alpha-level, and is denoted by the . Typical values for are: That is, rarely will we choose a very small or considerably large alpha-level. Suppose that we reject when the probability of observing something as or more extreme as what we have observed is 5% (or smaller). We have that . This means that there is still a 5% (or smaller) chance that we observe a value (sample mean, sample proportion, etc.) more extreme than what we have observed. That is, there is a 5% chance that we have falsely rejected the null hypothesis. Probabilistically, ( ) ( ) ( ) To visualize this, consider the diagram below. Recall that a conditional probability statement limits us to the event after the “pipe,” |, and then asks the question, “what percentage of the time can we expect the event to occur, out of the times the specified condition occurs. The modified table below shows that. Truth Don‟t Reject True True Positive 95% Reject False Negative (Type I Error) 5% 100% At this point we might wonder: why shouldn‟t we set Type 1 error risk extremely small so that we minimize the Good question. Imagine that your alpha is 0.0001. This means you will only reject 0.01% (or 1 out of 10,000 times) of the time, when it is true. Certainly, your risk of a Type I error is extremely small. Statistics for Decision-Making in Business © Milos Podmanik Page 214
Hypothesis Test Conclusion Since your decision criteria, or the numerical figure that we later calculate to decide whether or not to reject, will be extremely stringent and difficult to achieve. If this is the case, then you almost never reject the null hypothesis! Okay, so if you very rarely reject the null hypothesis, then you are also potentially committing another act of error: not rejecting the null hypothesis, even though it may be false. That is, you increase the likelihood of a Type II error. Recall that, ( ) ( ) We can see here that failing to reject results in potentially failing to reject it even when it should be rejected! Unfortunately, there is no free lunch in hypothesis testing. Truth Don‟t Reject True False Negative (Type II Error) Reject True Positive Though we cannot yet easily provide numerical support for this claim (which certainly makes sense), we will make the following preliminary conclusion: Type II Error - The probability of a Type II error, denoted , is inversely proportional to , the probability of a Type I error. That is, decreasing will increase . Important Caution Students are often confused that the probability of rejecting when is true and the probability of failing to reject when is true sum to 1. After all, these two possibilities are only two of the four possible results in a test decision. However, keep in mind that these are the percentages of time we reject and fail to reject out of all the times that is true! This out of only one column total, not the entire sample space. Statistics for Decision-Making in Business © Milos Podmanik Page 215
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Statistics for Decision- Making in
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A Note to Students This book is far
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6.3 Confidence Interval for ̂ 202
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which has undergone no additional r
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of transporting such an animal. Not
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Although far from last, we will con
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Account Type Revenue ($) New $5,296
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For the present time, we‟ll proce
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Row Labels Average of Revenue ($) M
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(SOURCE: Essentials of Modern Busin
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1.3Statistics in Excel When conduct
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We can make it a bit more presentab
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For measures such as the range, Exc
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We get: We can do the same for Old.
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Chapter 2 Visual Representations of
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We now have to enter a formula for
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We know from the data that this val
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Alternatively, it is possible to in
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16 14 12 10 8 6 Series1 4 2 0 SD D
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Selected the copied graph. In Chart
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2. The following data represents pe
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Frequency 2.2 Visualizing Quantitat
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You can either choose to have Excel
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If we had additional variables, the
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Select PivotChart and select the fi
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Frequency To make solid black lines
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Relative Frequency Histogram of Cal
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a. Using Excel, find the mean and r
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̅ ̅ account for this discrepancy,
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2.3.2 Percentile Another useful too
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2.3.4 Rank What if, on the other ha
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Check the “Analysis ToolPak” an
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You‟ll immediately notice that a
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̅ 2.4 Descriptive Statistics - Var
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√ ∑( ̅) √ √ This is what w
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This gives us a nice measure of how
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We can immediately see the mean and
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Lastly, a normal curve is the most
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Frequency ̅ Histogram of TV's Owne
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58.3 34.6 35.5 45.4 38.6 63.8 53.9
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Chapter 3 Probability and Decision
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Proportion of Rainy Days 1.2 Propor
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This means that if samples of HFCS
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Are we confident in accusing a lung
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3.2 Joint Probability In the previo
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Company 1 Choices Y Y Y Y Y Y Y Y N
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There is less than a 1% chance that
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We can now see that the situation i
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sequence, one ball is chose at rand
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3.3 Probability of Unions Imagine t
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Physical .20 .10 .30 Mental .40 .30
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SOLUTION: We first arrange this inf
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Homework Problems - 3.3 1. A gaming
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3.4 Conditional Probability In many
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) : The Arizona Cardinals make it t
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We could just as well have written,
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That is, the probability that the c
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This can happen in one of two ways:
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f. All numerical red cards are remo
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3.5 Combinations and Permutations R
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1 st Bushel 2 nd Bushel U1 U1 U2 U2
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Which, in its final state gives: Th
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Combination - Order Does NOT Matter
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( ) There is only a .9% chance that
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1. Are repeats/replacements allowed
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Object 4 Object 5 Also notice that
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e. 2. Your classmate was absent whe
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3.6 Expected Value Imagine that you
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An expected value is actually not s
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Where The expected value is: , - (
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SOLUTION: We first note that there
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Probability 0.7 0.6 0.5 0.4 0.3 0.2
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Statistics for Decision-Making in B
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For men, we see that the most frequ
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So, we had 10 trials and wanted to
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Example 1: A fair-two sided coin is
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( ) ( ) ( ) ( ) Example 2: A fair,
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Expected Value of a Binomial Random
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f. What is the probability that g.
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Probability For instance, we see th
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Density 0.25 Time Speng Waiting in
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Page 169 and 170: a. What is the probability that the
Page 171 and 172: Statistics for Decision-Making in B
Page 173 and 174: As you can see, this is a difficult
Page 175 and 176: We wish to know the area of the sha
Page 177 and 178: We can easily find that the probabi
Page 179 and 180: In the second section, we can check
Page 181 and 182: Chapter 6 Sampling Distributions an
Page 183 and 184: Thus, the standard deviation would
Page 185 and 186: 1.7 to 1.8 1.8 to 1.9 1.9 to 2 2 to
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Page 189 and 190: The Central Limit Theorem (CLT) has
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Page 195 and 196: Thus, we need to find the lower and
Page 197 and 198: We see that the standard deviation
Page 199 and 200: √ The lower limit is: √ And the
Page 201 and 202: following data was collected on the
Page 203 and 204: { So, we have a set of twenty 1‟s
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Page 207 and 208: Homework Problems -6.2 1. In a samp
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Page 213: Hypothesis Test Conclusion Test Say
Page 217 and 218: SOLUTION: d. A generic conclusion s
Page 219 and 220: 9. Based on the “Structure of a H
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Page 223 and 224: Relative Frequency 35% 30% 25% 20%
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Page 229 and 230: el freq Nitrous Oxide (thous. Tons)
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Page 235 and 236: 2. a. The long-run proportion of al
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Page 243 and 244: ̅ We have that ̅ and √ . So our
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