Statistics for Decision- Making in Business - Maricopa Community ...

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whether or not we have captured the true mean in our interval. On the upside, there is a probability associated with it! As a final note, it is interesting that we actually missed the true mean in our 95% confidence interval, since there is only a 5% chance of error. Keep in mind, however, that this interval was based on simulation. It is based on 1,000 samples and may have been better to increase the number of samples. 6.2.2 Confidence Interval for ̅ Using Theoretical Results – When and are Unkown In the previous section, we found that the sampling distribution of ̅ with is approximately normal with , ̅- and , ̅- . As a bit of notation, if a random variable has a normal distribution with mean and standard deviation, we would write: √ ̅ ( √ ) This reads, “ -bar is normally distributed with mean and standard deviation √ .” This, however, assumes that we know something that we probably don‟t – the population mean and standard deviation! As you might guess, we will use ̅ and √ to approximate these. This proposes a problem: we are introducing more error. In order to account for this, the normal distribution is not appropriate. When using these approximations, we must use the theoretical Student’s Distribution. This distribution looks much like the normal distribution, but is constructed by sample size, not the mean and standard deviation. Below is a comparison of the -distribution in comparison to the standard normal distribution for size . Statistics for Decision-Making in Business © Milos Podmanik Page 196
We see that the standard deviation (in red) is just slightly larger than that of the standard normal (in blue) – it is about 1.0339. So, as sample size gets greater, the -distribution begins to look more like a standard normal. BUT, look at the one below where sample size is 10: The variability is nearly 14% greater. As we mentioned, this distribution‟s shape relies on the sample size. The relationship is called the degrees of freedom and can be calculated as , that is degrees of freedom is equal to one less than the sample size. So, in our previous example, we had a sample size of 30, so In a probability calculator, we would enter 29 for the degrees of freedom: Statistics for Decision-Making in Business © Milos Podmanik Page 197
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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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Page 147 and 148: For men, we see that the most frequ
Page 149 and 150: So, we had 10 trials and wanted to
Page 151 and 152: Example 1: A fair-two sided coin is
Page 153 and 154: ( ) ( ) ( ) ( ) Example 2: A fair,
Page 155 and 156: Expected Value of a Binomial Random
Page 157 and 158: f. What is the probability that g.
Page 159 and 160: Probability For instance, we see th
Page 161 and 162: Density 0.25 Time Speng Waiting in
Page 163 and 164: Suppose we wish to find ( ), that i
Page 165 and 166: Density Resulting in a standard dev
Page 167 and 168: Density Without going into detail h
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
Page 187 and 188: 2.4 to 2.5 2.5 to 2.6 2.6 to 2.7 2.
Page 189 and 190: The Central Limit Theorem (CLT) has
Page 191 and 192: can set this up in our applet by ha
Page 193 and 194: 3) We cannot use this sample to cal
Page 195: Thus, we need to find the lower and
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
Page 205 and 206: DULY CAUTIONED: The assumptions her
Page 207 and 208: Homework Problems -6.2 1. In a samp
Page 209 and 210: Since this is a mathematical questi
Page 211 and 212: Example 3: Many older homes have el
Page 213 and 214: Hypothesis Test Conclusion Test Say
Page 215 and 216: Hypothesis Test Conclusion Since yo
Page 217 and 218: SOLUTION: d. A generic conclusion s
Page 219 and 220: 9. Based on the “Structure of a H
Page 221 and 222: 1.2 Descriptive VS. Inferential Sta
Page 223 and 224: Relative Frequency 35% 30% 25% 20%
Page 225 and 226: Symmetric: 35 30 25 20 15 10 5 0 10
Page 227 and 228: Repair Cost Mean 971 Standard Error
Page 229 and 230: el freq Nitrous Oxide (thous. Tons)
Page 231 and 232: Probability Pizza Size Distribution
Page 233 and 234: e. ( ) f. ; the average response ti
Page 235 and 236: 2. a. The long-run proportion of al
Page 237 and 238: e. 20% of all children are expected
Page 239 and 240: e. The middle 50% score between abo
Page 241 and 242: 5. b. This might indicate that the
Page 243 and 244: ̅ We have that ̅ and √ . So our
Page 245 and 246: 6. This is the probability that we
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