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Is that true at our college? You can ask the registrar for a copy of the SAT scores of a class (no names of course). Nationally females tend to have higher Verbal scores than males. Is there a significant difference between Verbal and Math scores for our students? 4. How do our SAT Math scores compare to those of another college? 3. What is the mean SAT-Math score at this college? 2. In these Tasks you will investigate four questions about SAT scores. SAT Performance The data for this are found at the end of this document. The second half of this task comes after Chapter 25, asking students to do two hypothesis tests – one involving paired data and the other comparing two independent means. And this affords you the opportunity to look at all the confidence intervals together, noting that most hit the target while others miss. Based on the various samples, students construct confidence intervals, then use the interval to compare the local performance to state or national results. The first part of the task asks students to select a random sample, reviewing use of random numbers and sampling issues from Chapter 12. SAT’s (or ACT’s) are ideal because each individual has two scores (making paired comparisons possible) and lots of statistics are available. Ideally you need a reasonably large but manageable data set 250 to 500 cases seems to work well. You may want to find something more relevant to your students and revise the task accordingly. We use the SAT scores reported for one college, but other data will work. The thrust of the tasks is more important than the particular data you use. This is the first of a pair of tasks that use the same data set. See Class Example 3.Ĭhapter 23 - Inferences About Means 393 Investigative Task Alternatively, we know it will take a sample about 9 times as large, 18(9) = 162 batteries, since the margin of error was decreased to a third of its size. We know that it’s going to take lots more batteries to cut the margin of error to a third of what it was. ⎛ s ⎞ ME = t * ⎜ ⎝ n ⎟⎠ ⎛ 29.31⎞ 15 = 2.145 ⎜ ⎝ n ⎟⎠ (2.145)(29.31) n= 15 n ≈ 17.56įinally, to estimate the mean battery lifespan to within 5 minutes, you could do the entire process again, perhaps using a critical value with much higher degrees of freedom. We would need to sample about 18 batteries in order to estimate the mean battery lifespan to within 15 minutes, with 95% confidence. Now, do a better n ≈ 14.67 estimate, using t14* = 2.145 as the critical value. (1.96)(29.31) n= 15 Our first estimate is about 15 batteries. ⎛ 29.31⎞ * 15 = 1.96 ⎜ First, do a preliminary estimate using z = 1.96 as the ⎝ n ⎟⎠ critical value. Sample Size – We want to know how many batteries to test ME = t * ⎛ s ⎞ ⎜⎝ ⎟ to be 95% sure of estimating the mean lifespan to within n⎠ 15 minutes. Confidence Interval – The conditions have been met, so we can create a one-sample t-interval, with 90% confidence.Ĭopyright © 2012 Pearson Education, Inc. It does not appear that the company has met its goal. There is no evidence to suggest that the mean battery lifespan exceeds 300 minutes. P-value = P( y > 306.25) = P(t11 > 0.7387) = 0.238Ĭonclusion – Since the P-value is high, fail to reject the null hypothesis. Since the conditions have been met, we can do a one sample t-test for the mean, with 11 degrees of freedom. Nearly Normal Condition: The distribution of battery lifespans is unimodal and symmetric, so it’s reasonable to assume that the lifespans of all batteries could be described by a Normal model. However, it is reasonable to assume that these batteries are representative of all batteries. Randomization Condition: This is not a random sample of batteries, but merely 12 batteries produced for preliminary testing. The alternative hypothesis is that the batteries have a mean lifespan greater than 300 minutes. Hypotheses – The null hypothesis is that the batteries have a mean lifespan of 300 minutes. We have 12 battery lifespans in our sample to test the claim. We want to know if the mean battery lifespan exceeds the 300-minute goal set by the manufacturer. Chapter 23 Inferences About Means Chapter 23 – Solutions to Class Examples 1.