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Q3a) Using a Confidence Interval to Test a Hypotheis

SOLUTION

This is an odd question that only rarely appears on exams. You are asked to determine if the mean of a population falls within a specific interval:

19.16mg ± 1 = 18.16mg to 20.16mg

For this reason, we cannot just use the t-test of an unknown population mean. Instead, we must calculate the 99% confidence interval (which deals with intervals), and then see how that interval compares to the claimed interval of 18.16mg to 20.13mg.

If the entire 99% confidence interval falls within this claimed range, it supports the hypothesis that the mean THC level is within 1 mg of 19.16 mg at the 1% significance level.

Given:

Sample mean: \bar{x} = 19.16 mg

Sample variance: s^2 = 0.0330

Sample size: n = 19

Calculations

99% Confidence Interval:

Critical t-value for 99% confidence level with 18 degrees of freedom: t_{\alpha/2} = \pm 2.878

Standard Deviation: s = \sqrt{0.0330} = 0.1817 mg

    \[ CI = \bar{x} \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}} \]

    \[ CI = 19.16 \pm 2.878 \times \frac{0.1817}{\sqrt{19}} \]

    \[ CI = 19.16 \pm 2.878 \times 0.0417 \]

    \[ CI = 19.16 \pm 0.1201 \]

Confidence Interval: 19.0399, 19.2801 mg

Conclusion

We estimate that the mean THC amount falls between 19.0399mg and 19.2801mg, and this type of estimator is correct 99% of the time

Since this interval lies entirely within the range of 18.16 mg to 20.16 mg, we can say there is sufficient evidence (at the 1% significance level) that GH’s product is within 1 mg of 19.16 mg.

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