A1 Garth Tarr C12345678 PDF

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Stat1070 Complete assignment 1...

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STAT1070 Assignment 1 Name: Student ID: 12345678 If an extension was granted, please attach a copy of your extension approval to your submission.

Question 1 (a) Because carat weight is a continuous variable, a 1-sample t-test is appropriate. Let μ be the true average carat weight of gold pieces listed as 18-carat gold. H0: μ = 18 HA: μ < 18 Test statistic: t = −1.205 (see t column in Figure 1) Null distribution: If H0 is true, the null distribution is a t distribution with 19 degrees of freedom, t19. The degrees of freedom parameter is equal to n-1 = 19 as there are n = 20 observations. p-value: P (t19 ≤ −1.205) = .1215 (dividing the 2-sided p-value in Figure 1 by 2) Conclusion: Because the p-value is relatively large, fail to reject H0. There is insufficient evidence to suggest that pieces listed as 18-carat gold have average carat weights less than 18.

Figure 1: SPSS output from Analyze → Compare Means → One-Sample T Test.

(b) In Figure 1, a 95% confidence interval for the difference between μ and 18 is [−0.82, 0.22], rounded to 2 decimal places. Hence, a 95% confidence interval for μ can be found by adding 18 to both limits, yielding an interval of [17.18, 18.22]. Thus we can be 95% confident that the average weight of gold pieces listed as 18-carat gold is somewhere between 17.18 carats and 18.22 carats. This interval includes the target 18 carats, which means that it is plausible that the true average carat weight for gold pieces listed as 18-carat gold really is 18 carats. Hence, this interval reinforces the decision to fail to reject the null hypothesis in part (a).

(c) Both the 1-sample t-test and the confidence interval for μ assume that (1), observations are independent within the sample and (2) observations come from a normal distribution OR the sample size is large enough so that you can rely on the central limit theorem. The independence assumption is reasonable because the 20 gold pieces in the sample were randomly selected. The normal quantile plot in Figure 2 does not suggest any serious departure from normality, as all observations lie reasonably close to the straight line. Hence, the assumption that the observations have been sampled from a normal distribution is satisfied.

Figure 2: Normal quantile plot of carat weights of the 20 gold pieces in the sample....