Ch. 8 Estimating with Confidence Ch. 8-1 Confidence Intervals: The Basics

confidence intervals significance tests confidence intervals significance tests Introducing confidence intervals visually. I need a volunteer.

current AP Stats students 𝜇 4 students 𝑥 Let’s take a sample of 4 and find 𝑥 Based on our sample, we estimate the mean exam score is ___

point estimator statistic parameter point estimate What does this remind you of? Ch. 7 sampling distribution Sampling Distribution of 𝑥 normal because pop. dist. is normal N(𝜇, 𝜎 𝑥 ) 𝜇 (unknown mean) = 10 4 =5 𝜎 𝑥 = 𝜎 𝑛 𝜇 10% condition 2 5 =10 𝑛≤ 1 10 𝑁 4≤ 1 10 40 10

______ 10 Write this every time!! ____ ±10= , REMEMBER THE WORDING!! ____ ±10= , REMEMBER THE WORDING!! We are 95% confident that the interval from _______ to _______ captures the actual value of the mean final exam score. context confident We don’t say 95% chance or probability of capturing the actual value of the parameter because the interval either does (prob = 1) or doesn’t capture it (prob = 0).

The confidence level 𝐶 tells us how likely it is that the method we are using will produce an interval that captures the population parameter if we use it many times. Doesn’t tell us the chance that a particular interval captures the parameter. A confidence interval gives a set of plausible values for the parameter. If we were to repeat the sampling procedure many times, about 95% of the confidence intervals computed would capture the mean final exam score. 𝑥 = ____ _____ ± 10= , 𝑥 = ____ _____ ± 10= , Use the Java applet to take many many samples of size 4 to make many many confidence intervals. Draw the picture to the right.

How much does the fat content of Brand X hot dogs vary How much does the fat content of Brand X hot dogs vary? To find out, researchers measured the fat content (in grams) of a random sample of 10 Brand X hot dogs. A 95% confidence interval for the population standard deviation σ is 2.84 to 7.55. Interpret the confidence interval. Interpret the confidence level. True or false: The interval from 2.84 to 7.55 has a 95% chance of containing the actual population standard deviation σ. Justify your answer. We are 95% confident that the interval from 2.84 to 7.55 g captures the true standard deviation of the fat content of Brand X hot dogs. If we were to repeat the sampling procedure many times, about 95% of the confidence intervals computed would capture the population standard deviation. False. The probability is either 1 (if the interval does contain the true st dev) or 0 (if it doesn’t).

Shorter interval, lower % hit Bigger interval, higher % hit Bigger interval, highest % hit (that you’ll be dealing with) increases increasing

Estimate ± (crit value)(std dev) Margin of Error depends on critical value and std dev. Estimate ± 2 𝜎 𝑥 95% confidence Estimate ± 3 𝜎 𝑥 99.7% confidence General Formula: Estimate ± (crit value)(std dev) The critical value depends on the confidence level. We will be calculating different critical values in Ch. 8-2 and 8-3. What kind of critical values would reduce the margin of error? lower values Lower standard deviation by _____________________ increasing sample size Larger samples give more precise estimates  _______________ less variability Why don’t we always get large sample sizes in the real world? Very costly and time-consuming

𝑝 𝜇 random randomized normal the population distribution is normal OR by the CLT if 𝑛≥30. 𝑛𝑝≥10 and 𝑛 1−𝑝 ≥10 10% condition

SRS stratified cluster random sampling random assignment