Statistics Ch.9
Estimate the lower bound of a confidence interval for a population proportion
Of course we will start with p-hat or the sample proportion - the margin of error, (we add the margin of error to get the upper bound, it's pretty much the critical value or z subscript alpha/2 times the standard deviation of the sampling distribution,in other the number (z score) of standard deviation away from the mean of the sampling proportion which is equal to the population proportion. (see formulas sheet in Documents)
Level of confidence; how is it expressed
Represents the expected proportion of confidence intervals that will contain the parameter (out of all the confidence intervals present0 if a large number of samples is obtained'; the proportion will then be expressed as %
Round up 11.2 is it 11 or 12, when do we round up in this chapter?
Rounding up means going to the next larger integer which in this case will be 12; we use rounding up when we estimate the sample size given a level of confidence and a certain margin of error (see formulas in Table)
What's the benefit of the boxplot and the normal points graph
The boxplot needs to be checked to ensure there are no outliers in the data because this will drastically affect our sample mean that we use to estimate the population mean!!!! . The normal graph curve tells us that all the points of the graph lie within a linear pattern between the lines (see h.w. problem #13) which assures us that the sample is normally distributed (especially when it is under 30)
95 % level of confidence refers to confidence in .... and not ....
The method; the interval. For instance we are confident that our method works for 95% of the samples. It doesn't tell us that the paramter is within the upper and lower bounds of the confidence interval - see what she says tomorrow.
Conditions to use confidence intervals
The sample is obtained is simple or less than or equal to .o5N The sample is obtained is random. The sampling distribution is normal (sample size is greater than or equal to 30 regardless of the distribution of the population OR the sample size can be less than 30 provided that the population is normally distributed).
Point Estimate
The value of a statistic that is used to estimate the value of a parameter. For instance the value of p-hat (the sample proportion) is used to estimate p or the population proportion.
Explain a 95 % level of confidence in colloquial terms
We're 95 percent confident that the population parameter is between the lower bound of the confidence interval and the upper bound of the confidence interval.Recall that confidence interval is a range.
What does a 95% level of condidence mean
1)That 95% of all confidence intervals are intervals that contain the population parameter. 2) In other words and more accurately, it means that 95 percent of the obtained samples (and hence sample proportions) of size n result in a confidence interval that INCLUDES THE POPULATION PARAMETER. OR 5 percent of the sample proportions of size n obtained resulted in confidence interval that does NOT include the parameter
Properties of t-Distribution
1. t-Distribution is different for different degrees of freedom 2. The distribution is centered and symmetric about 0; right half = left half = 1/2 3. Area under the curve is 1, Read other properties of pg. 443 as necessary (didn't understand it)
Explain what confidence of interval of 95% means given a sample of size n.
95% level of confidence means that if we took all possible samples of size n from the population of an unknown parameter, 95% of all the possible samples of size n that can be randomly obtained will result in a confidence interval that contains the population parameter
Format (formula) of a confidence interval
A confidence Interval will estimate the population parameter in this format: Point Estimate ± margin of error; Of course as you increase the margin of error, you increase the level of confidence that you have in your estimate.
A confidence Interval of an unknown parameter
A range of numbers such as from 22-30 BASED ON POINT ESTIMATE that is used to estimate an unknown parameter
What's the idea behind an interval for estimating population parameters. Consider using Peoples' usage of Facebook as an example; Discuss Intervals and Confidence.
Consider obtaining a sample proportion of .48 or 48% people use Facebook and then you take another sample and its proportion happens to 50%, to solve this mystery you say make an interval (or range) of values that you say will include the TRUE percentage or proportion of people who use Facebook. Of course the more you increase the interval the more confident you are that it includes the real percentage or proportion of individuals in the population who use Facebook.
Review from previous chapter: Conditions for normal distribution of sampling proportions or sample mean
For the sampling distribution to be normally distributed, np(1-p) needs to be larger than or equal to 10 and the sample size needs to be less than or equal to .05N (population size) and if n is greater than or equal to 30 then distribution is approximately normal.
Conditions for using the confidence interval
If you look at the formulas for Ch.9, you'll notice that we use test statistics (such as z or t scores) and hence we'll be looking at a normal distribution; hence sample size should be less than or equal to .05N, this will allow for the second condition which is the independent trials. Also we want to have a normal distribution to use the confidence interval so n should be greater than 30 and np(1-p) is greater than or equal to 10.
Prescene of outliers in the data
Increase the width of the interval in the data see q15 part c.
How does t-statistic help us?
Like the z score, it gives us the NUMBER of errors that was used in order for us to estimate how much is the sample mean away from the population mean.
Very important note about confidence interval; what does 99 percent confidence level mean again (3 things).
Notice that a confidence interval is really representing a certain population proportion p-hat with a margin of error, that's why it is an interval. Hence a 99% confidence level means that 99 percent of all confidence intervals contain the population proportion or 99 percent of all samples or sample proportions will give you a confidence interval that contains the population proportion or we're 99 confident that the confidence interval contains the population proportion
For sections 9.1 and 9.2: Make sure you look at formulas sheet that I saved in documents section because were not included here, make sure you know how to use them.
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Caution a confidence interval of 95 percent for instance does NOT mean
that the probability that the parameter is present in a certain interval is 95%. Recall probability describes the likelihood of an event that hasn't already been determined. In this chapter, there is a certain value in the population that for instance use Facebook, we use the confidence interval as a range of values based on the point estimate to estimate this already determined population proportion. See pg 429 if necessary.
As we increase the sample size, we decrease .... and hence we decrease also
the margin of error; level of confidence.
