This covers sections 8.1, 8.2, and 8.3. In section 8.3, we will not be covering how to deal with Small Samples. We will NOT cover 8.4. 8.5 is a review, so it’s worth reading. Next week, we will move on to chapter 9. Homework: Section 8.1: 43, 44, 49, 50 Section 8.2: 29, 30, 35, 36 Section 8.3: 27, 28 1) A sample of size n = 80 is drawn from a population whose standard deviation is σ = 6.8. The sample mean is 𝑥̅ = 40.4. Construct a 90% confidence interval for µ Does the population need to be distributed normally for the confidence interval to be valid? 2) Look at the z-interval menu on the calculator. Look at the t-interval menu on the calculator. How are they different (using either Stats or Data mode)? So what is the difference between a Z-interval and a t-interval? -----------------------A study is done to discover the average amount of money that households donate to non-profits. Assumption: distribution of donations is approximately normal (our sample size is not >30) so the distribution of the mean is approximately normal. After surveying 25 households, a sample mean of $732 was found. The population standard deviation is known to be $120. Using the calculator, construct a 95% confidence interval for the mean donation. What’s the standard error for the sample mean? What’s the critical zα/2 for a 95% confidence interval? What’s the margin of error for a 95% confidence interval? We would like to have a margin of error be +/- of $10 (at 95% confidence). How many household would need to be sampled to have a margin of error be $10? z n /2 m 2 Problem is same as above, but population standard deviation is not known Assumption: distribution of donations is approximately normal (our sample size is not >30) so the distribution of the mean is approximately normal. After surveying 25 households, a sample mean of $732 was found. The standard deviation of the sample is $120 Since we don’t have the population standard deviation and must use a sample, we will not use a normal distribution and z-scores. Instead we will use which distribution? Construct a 95% confidence interval for the mean donation. How many degrees of freedom are there? What’s the estimated standard error for the sample mean? Arguments for the inverted t-distribution is invT(area in left tail, degrees freedom) What’s the critical tα/2 for a 95% confidence interval? What’s the margin of error for a 95% confidence interval? Note: when sample size gets large (>200), the t-distribution becomes very close to the normal distribution, and some people will just switch to normal for critical values. However, one should still refer to the distribution as a t-distribution if the population standard deviation is not known. --------------------------Proportions: A survey of 150 adults found that 38% of those surveyed bought music over the internet. What are the conditions necessary to construct a confidence interval for a population proportion? Construct a 98% confidence interval for the population mean. What’s the standard error for the proportion? What’s the critical zα/2 for a 98% confidence interval? What’s the margin of error for a 98% confidence interval? If the music industry wants to conduct another survey, and they want the margin of error to be 1% point, how many adults will need to be surveyed (still 98% confidence)? z If there’s an estimate for p̂ : n pˆ (1 pˆ ) / 2 E 2 z If not: n .25 / 2 E 2