Confidence Intervals and Hypothesis Tests: Two Samples - Florida ...

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Example 142 Use the data from the FCAT problem that opened this section to form a 90% confidence interval for µ d the true average differe difference between before and after FCAT scores and interpret inter the results. STATSprofessor.com Chapter 9 9.9 Inference Procedures to Compare Two Population Proportions: Independent Sampling Many times we have a need to answer questions that involve the comparison of two population proportions. Consider the example below: Example 143: A product reliability study was conducted on laptops by the company Square Trade Inc. Among 11,500 entry-level level laptops (priced between $400 - $1000) followed for one year, 541 malfunctioned. Among 9,300 premium laptops (priced over $1000) followed for one year, 391 malfunctioned. Use a 0.01 significance level to test the claim that the incidence of mal malfunction mal is higher for entry-level laptops than it is for premium laptops. Does es it seem that spending more money on a laptop will offer you some protection otection from the inconvenience of a malfunction? 18
STATSprofessor.com Chapter 9 To answer this question we need to know what quantities to compare. Let’s look at what we have here: Entry-level Premium X 541 391 n 11,500 9,300 ˆp 0.047 0.042 Clearly, for this sample, premium laptops had a slightly lower malfunction rate for the first year, but is this difference just a coincidence? Maybe this could have happened by chance—after all the there is always some random sampling error present. What we can do is create a test statistic around the difference between these two sample proportions ( pˆ pˆ ) − using the standard deviation of the sampling distribution for the difference between these 1 2 two sample proportions. Then if the sample sizes are large enough*, we can express the probability that we would observe this difference ( pˆ pˆ ) assumption under the null hypothesis). − by random chance given that they are really the same (the 1 2 *Sample sizes are large enough when ˆ 3 ˆ ˆ pq p ± is entirely captured inside [0, 1]. Another rule of n thumb that is used is: Samples sizes are large enough if n1 pˆ ˆ 1 ≥ 15 and n1q1 ≥15 n pˆ ≥15 and n qˆ ≥15 So our point estimator for the true population difference will be: ( pˆ − pˆ ) The standard error of its sampling distribution will be: p q p q ⎛ 1 1 ⎞ x + x + ≈ pq ˆ ˆ ⎜ + ⎟, where pˆ = n n ⎝ n n ⎠ n + n 1 1 2 2 1 2 1 2 1 2 1 2 2 2 2 2 1 2 19
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