#  With replacement or without replacement?  Draw conclusions about a population based on data about a sample.  Ask questions about a number which describe.

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 With replacement or without replacement?

 Draw conclusions about a population based on data about a sample.  Ask questions about a number which describe a population.  Numbers which describe populations are called…  Parameters  To estimate a parameter, choose a sample from the population and use a statistic (a number calculated from the sample).

 Population Proportions: p  Sample Proportions:  (also known as p-hat) – we’ve seen this before… it’s the count in the sample which applied (i.e., said yes) divided by the sample size  Example: The 2001 Youth Risk Behavioral Survey questioned a nationally representative sample of 12,960 students in grade 9-12. Of these, 3340 said they had smoked cigarettes at least one day in the past month.  Who is the population?  How do you write the sample proportion?

 The population is high school students in the United States.  The sample is the 12,960 students surveyed.  The sample proportion (p-hat) …  Give all answers to 4 decimal places.

 The parameter is the proportion that have smoked cigarettes in the past month. (This is usually an unknown value).  Using the value of p-hat, we can make generalized statements about the population.  Based on this sample, we can estimate that the proportion of all high school students who have smoked cigarettes in the past month is about 25.77%.

 On the last example, the answer was about 25.77%. In order to capture the true population parameter in 95% of all samples, use a 95% confidence interval…  Remember that with a 68-95-99.7 curve, 95% is 2 standard deviations in each direction.  This is the formula we will use…

 Using this formula will give two endpoints for a confidence interval. 95% of all samples will fall in this interval.  In this formula, the value under the square root represents the standard deviation. n stands for the number in the sample.

 Remember, p-hat=0.2577 and n=12960.

 This interval catches the true unknown population proportion in 95% of all samples.  In other words, we are 95% confident that the true proportion of high school students who have smoked cigarettes at least one day in the past month is between 25.01% and 26.53%.

 Estimate ± Margin of Error  What would the confidence interval be here?

 So far, we have used 68% = 1 std. dev., 95% = 2 std. dev., and 99.7% = 3 std. dev.  These were estimates which were used.  More accurate values are Critical Values, denoted z* (this is on page 502):

 Z* now takes the place of the number of standard deviations outside the square root.

WWe are 95% confident that the true proportion of high school students who have smoked cigarettes at least one day in the past month is between 25.03% and 26.51%. EEarlier, with using “2” for the number of standard deviations, instead of “1.96” as the critical value z*, we had stated: WWe are 95% confident that the true proportion of high school students who have smoked cigarettes at least one day in the past month is between 25.01% and 26.53%.

 While this is a very small difference, the benefits with using critical values is that there are more options to use (8) than with the 68-95-99.7 rule’s 3 standard deviations.  Critical Values z* can be used for the following confidence levels: 50%, 60%, 70%, 80%, 90%, 95%, 99%, and 99.9%.  Standard Deviations can be used for 68%, 95%, and 99.7%

 Remember, the larger the number, the wider the interval…if you don’t need a high proportion of confidence, you can use a smaller number (like 50%). 50% will give a smaller confidence interval (will contain a smaller proportion of the true unknown population).  50%: (0.2551, 0.2603)  60%: (0.2545, 0.2609)  70%: (0.2537, 0.2617)  80%: (0.2528, 0.2626)  90%: (0.2514, 0.2640)  95%: (0.2503, 0.2651)  99%: (0.2478, 0.2676)  99.9%: (0.2451, 0.2704)  So the smallest interval is 25.51% to 26.03% and the largest interval is 24.51%-27.04%

 What is the formula for p-hat?

 What is the formula for a confidence interval, using z* (just the basic formula—no numbers)?

 What does n stand for in the equation?  The sample size.

 How many critical values (z* values) are there to choose from?  8…they are 50%, 60%, 70%, 80%, 90%, 95%, 99%, and 99.9%

 Page 495-496, #9.1-9.5  Page 505, #9.9, 9.10