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Let's review confidence intervals When population standard deviation is known,
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we use confidence intervals when we don't know the population average
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It gives us a range of possible sample averages and tells us the percentage confidence that
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the population average falls within.
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In our classroom we took a poll of five students in the class and
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found that the mean age in our sample is 27.
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We know our population.
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Standard deviation is six.
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What is your range for each of the following, construct a confidence interval
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that the population means is likely to fall into So
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in 95% confidence
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What would our range be?
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What would our range be?
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We need to use our formula
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x bar plus minus
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z a/2 the population standard deviation
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z a/2 the population standard deviation
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Here our X bar is equal to the sample average
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Here our X bar is equal to the sample average
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said a divided by two is equal to the Z score
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of the confidence interval
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is the population
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standard deviation
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and our sample size.
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So let's enter this in here
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we have 27
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plus minus.
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Now the zed score we can use our
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probability table
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to show a 95% confidence
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or we're looking for is a 0.9750 So what this would
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mean would be at a 5% less.
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We would have 2.5% on either side of it being either
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over or under.
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So these air pretty standard.
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You can also find the confidence intervals on your formula sheet
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here so we can use 1.96
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Could you better explain it?
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We assume that we have a 25 for me.
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A 95% confidence that would leave a 2.5% chance to population mean will be
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lower.
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Then I arrange,
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and a 2.5% a chance that
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the we will be higher than arranged.
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So negative 1.96 and 1.96 actually equal to a
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95% confidence rate.
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1.96
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Our population center deviation is six,
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divided by our sample of five.
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This equals 27 plus minus
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five points 26 So we can
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say with 95% confidence rate.
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That the age will be the mean will be 27 plus minus
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5.26 or
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21.74
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2 32 point 26
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So within 90% Confidence
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27 plus minus.
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Here in our formula sheet 1.645
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we have a less of a confidence rate.
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Meaning we are okay with being 10% wrong here,
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we're gonna have our Z score of 1.645
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Modified six over square to five.
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That gives us the confidence interval of 27
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plus minus four points.
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41 So that is less than the last one.
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The range on this
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would be a 22.59
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to 31.41 So,
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as you can see,
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our total range is
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smaller than a range here.
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So we have less confidence that this is correct.
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So we have a smaller range.
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When we need more confidence.
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We have a larger range.
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So let's have a look.
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What happens with 99% confidence
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27 plus minus,
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we can use the zed score.
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2.576
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six,
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divided by a squared of five equals
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27 plus minus
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6.91
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So a range is 20.09 to 33.91
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With 99% confidence,
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With 99% confidence,
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we can be sure that the average age in our class is 20 years and
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20.09 to 33.91 So much larger range for a higher
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level of confidence,
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now at 95% confidence.
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But with the sample size of 16 what will happen?
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27 plus minus
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1.96
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Standard deviation of 65 divided by the square root of 16
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we get a confidence of 27 plus minus
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2.94 or a range
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of 24.06
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of 24.06
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to 29.94
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29.94
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as you can see with a larger sample size,
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when we get a better representation of the population,
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it actually makes a range smaller because we have more information.
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If 95% confidence with the sample size of 16
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we have a plus minus 2.94 a range of about six
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with 95% confidence,
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but a sample of five.
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We have a range 27 plus minus five and a quarter with a range
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of about 10.5,
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so the larger the sample size,
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or the less confidence you have,
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We'll bring down the confidence interval.
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We'll bring down the confidence interval.
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We'll bring down the confidence interval.
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We'll bring down the range that is used