STAT 3360 Notes

• Midpoint: \(\boxed{ \bar{X} }\);
• Critical Value: \(\boxed{ Z_{\alpha/2} }\);
• Margin of Error: \(\boxed{ Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} }\);
• Lower Confidence Limit (LCL): \(\boxed{ \bar{X} - Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} }\);
• Upper Confidence Limit (UCL): \(\boxed{ \bar{X} + Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} }\);
• Confidence Interval: \(\boxed{ [\bar{X} - Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}, \bar{X} + Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}] }\);
• Width of Confidence Interval: \(\boxed{ 2 \cdot Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} }\).

• the Z-test for \(\boxed{H_0: \mu \le \mu_0 \text{ vs } H_A: \mu > \mu_0}\) (which is equivalent to \(\boxed{H_0: \mu = \mu_0 \text{ vs } H_A: \mu > \mu_0}\)) can be constructed as follows,
  • Test Statistic: \(\boxed{T = \frac{\sqrt{n}(\bar{X} - \mu_0)}{\sigma}}\);
  • Critical Value: \(\boxed{Z_{\alpha}}\);
  • Rejection Rule: Reject \(H_0\) if \(\boxed{T > Z_{\alpha}}\).
• the Z-test for \(\boxed{H_0: \mu \ge \mu_0 \text{ vs } H_A: \mu < \mu_0}\) (which is equivalent to \(\boxed{H_0: \mu = \mu_0 \text{ vs } H_A: \mu < \mu_0}\)) can be constructed as follows,
  • Test Statistic: \(\boxed{T = \frac{\sqrt{n}(\bar{X} - \mu_0)}{\sigma}}\);
  • Critical Value: \(Z_{1 - \alpha}\), which equals \(\boxed{- Z_{\alpha}}\);
  • Rejection Rule: Reject \(H_0\) if \(T < Z_{1 - \alpha}\), which is equivalent to \(\boxed{T < -Z_{\alpha}}\).
• the Z-test for \(\boxed{H_0: \mu = \mu_0 \text{ vs } H_A: \mu \neq \mu_0}\) can be constructed as follows,
  • Test Statistic: \(\boxed{T = \frac{\sqrt{n}(\bar{X} - \mu_0)}{\sigma}}\);
  • Critical Values: \(Z_{1 - \alpha/2}\) (which equals \(\boxed{- Z_{\alpha/2}}\)) and \(\boxed{Z_{\alpha/2}}\);
  • Rejection Rule: Reject \(H_0\) if \(T > Z_{\alpha/2}\) or \(T < - Z_{\alpha/2}\), which is equivalent to \(\boxed{|T| > Z_{\alpha/2}}\).

• Let \(X\) be the score of a student, then we know
  • \(X\) follows a Normal distribution;
  • the population mean of \(X\), denoted by \(\mu\), is unknown;
  • the population standard deviation of \(X\) is \(\sigma = 3\);
  • the sample size \(n = 16\);
  • the sample mean \(\bar{X} = 79\);
  • the sample standard deviation is \(s = 3.5\).
• To construct the confidence interval of \(\mu\) when population standard deviation \(\sigma = 3\) is known,
  • the confidence level is \(1 - \alpha = 90\%\), so \(\alpha = 0.1\);
  • the midpoint of the confidence interval is \(\bar{X} = 79\);
  • the critical value is \(Z_{\alpha/2} = Z_{0.1/2} = Z_{0.05} = 1.64\) because \(P(Z > 1.64) = 0.05\);
  • the margin of error is \(Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} = 1.64 \cdot \frac{3}{\sqrt{16}} = 1.23\);
  • the lower confidence limit is \(79 - 1.23 = 77.77\);
  • the upper confidence limit is \(79 + 1.23 = 80.23\);
  • the confidence interval is \([77.77, 80.23]\);
  • the width of the confidence interval is \(2 \times 1.23 = 2.46\).

• the confidence level is \(1 - \alpha = 90\%\), so \(\alpha = 0.1\);
• the critical value is \(Z_{\alpha/2} = Z_{0.1/2} = Z_{0.05} = 1.64\) because \(P(Z > 1.64) = 0.05\);
• since the width of the confidence interval is \(2M\) where \(M\) is the margin of error, to let \(2M < 2\), we need \(M < 1\);
• by formula, to achieve \(M<1\), the smallest sample size should be \(n = \left( \frac{z_{\alpha/2} \sigma}{M} \right)^2 = \left( \frac{1.64 \cdot 3}{1} \right)^2 = 24.21 \approx 25\).

• Here we want to confirm the claim that \(\mu < 80\), so we let it be the alternative hypothesis. That is, we want to test \(H_0: \mu \ge 80\) vs \(H_A: \mu < 80\) (or equivalently \(H_0: \mu = 80\) vs \(H_A: \mu < 80\));
• The test statistic is \(T = \frac{\sqrt{n}(\bar{X} - \mu_0)}{\sigma} = \frac{\sqrt{16}(79 - 80)}{3} = -1.33\);
• The significance level is \(\alpha = 5\%\);
• The critical value is \(- Z_{\alpha} = - Z_{0.05} = - 1.64\);
• The rejection rule is: reject \(H_0\) if \(T < - Z_{\alpha}\);
• Here \(T = -1.33 > -1.64 = -Z_{\alpha}\), so we don't have enough evidence to reject \(H_0\) and thus don't have enough evidence to say that the population mean is lower than \(80\).

• The hypotheses are the same, ie, \(H_0: \mu \ge 80\) vs \(H_A: \mu < 80\) (or equivalently \(H_0: \mu = 80\) vs \(H_A: \mu < 80\));
• The test statistic is the same, ie, \(T = \frac{\sqrt{n}(\bar{X} - \mu_0)}{\sigma} = \frac{\sqrt{16}(79 - 80)}{3} = -1.33\);
• The significance level now is \(\alpha = 10\%\);
• The critical value now is \(- Z_{\alpha} = - Z_{0.1} = - 1.28\);
• The rejection rule is the same, ie, reject \(H_0\) if \(T < - Z_{\alpha}\);
• Here \(T = -1.33 < -1.28 = -Z_{\alpha}\), so we have enough evidence to reject \(H_0\) and thus have enough evidence to say that the population mean is lower than \(80\).

• Comparing Q4 with Q3, we see that at a higher significance level, we are more likely to reject \(H_0\). Recall that the significance level is just the probability of Type I error, and Type I error occurs if we reject \(H_0\) while \(H_0\) is true. With a higher significance level, we are allowed more chance for Type I error and thus are more aggressive in rejecting \(H_0\).
• In Q3, we were conservative (prudent) in rejecting \(H_0\) by using a lower significance level \(\alpha = 5\%\). We didn't reject \(H_0\) and thus avoided the risk of Type I error. However, if \(H_0\) is not true, then we miss the chance to disprove it (ie, Type II error occurs).
• In Q4, we were aggressive in rejecting \(H_0\) by using a high significance level \(\alpha = 10\%\). We rejected \(H_0\) and supported \(H_A\). However, if \(H_0\) is true and \(H_A\) is not true, then Type I error occurs.
• Therefore, the choice of significance is critical for the decision making, and different significance level can lead to even opposite conclusions for the same data.
• In general, a lower significance level means a conservative (prudent) attitude toward Type I error, which leads to a low chance of rejecting \(H_0\). Therefore, at a low significance level,
  • if \(H_0\) is rejected, then we believe we have enough evidence to support \(H_A\) because we are already prudent;
  • if \(H_0\) is not rejected, then we don't say "\(H_0\) is true". Instead, we say "we don't have enough evidence to reject \(H_0\)". When we are conservative, we need very strong evidence for rejecting \(H_0\). Therefore, it may happen that \(H_0\) is not true but we choose not to reject it because we believe the evidence is not strong enough.
• In general, a high significance level means an aggressive attitude toward Type I error, which leads to a high chance of rejecting \(H_0\). Therefore, at a high significance level,
  • even if \(H_0\) is rejected, we are still not quite confident in \(H_A\) because we are inclined to reject \(H_0\) and thus have a high chance of rejecting a true \(H_0\);
  • if \(H_0\) is not rejected, then we believe there is enough evidence to support \(H_0\) (ie, \(H_0\) is not rejected even when we are aggressive).

• For example, the number in the red box means \(P(t_{15} > 1.753) = 0.05\), so \(t_{0.05, 15} = 1.753\).

  

• Midpoint: \(\boxed{ \bar{X} }\);
• Critical Value: \(\boxed{ t_{\alpha/2, n-1} }\);
• Margin of Error: \(\boxed{ t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}} }\);
• Lower Confidence Limit (LCL): \(\boxed{ \bar{X} - t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}} }\);
• Upper Confidence Limit (UCL): \(\boxed{ \bar{X} + t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}} }\);
• Confidence Interval: \(\boxed{ \left[ \bar{X} - t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}, \bar{X} + t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}} \right] }\);
• Width of Confidence Interval: \(\boxed{ 2 \cdot t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}} }\).

• the t-test for \(\boxed{H_0: \mu \le \mu_0 \text{ vs } H_A: \mu > \mu_0}\) (which is equivalent to \(\boxed{H_0: \mu = \mu_0 \text{ vs } H_A: \mu > \mu_0}\)) can be constructed as follows,
  • Test Statistic: \(\boxed{T = \frac{\sqrt{n}(\bar{X} - \mu_0)}{s}}\);
  • Critical Value: \(\boxed{t_{\alpha, n-1}}\);
  • Rejection Rule: Reject \(H_0\) if \(\boxed{T > t_{\alpha, n-1}}\).
• the t-test for \(\boxed{H_0: \mu \ge \mu_0 \text{ vs } H_A: \mu < \mu_0}\) (which is equivalent to \(\boxed{H_0: \mu = \mu_0 \text{ vs } H_A: \mu < \mu_0}\)) can be constructed as follows,
  • Test Statistic: \(\boxed{T = \frac{\sqrt{n}(\bar{X} - \mu_0)}{s}}\);
  • Critical Value: \(t_{1-\alpha, n-1}\), which equals \(\boxed{- t_{\alpha, n-1}}\)
  • Rejection Rule: Reject \(H_0\) if \(T < t_{1-\alpha, n-1}\), which is equivalent to \(\boxed{T < - t_{\alpha, n-1}}\)
• the t-test for \(\boxed{H_0: \mu = \mu_0 \text{ vs } H_A: \mu \neq \mu_0}\) can be constructed as follows,
  • Test Statistic: \(\boxed{T = \frac{\sqrt{n}(\bar{X} - \mu_0)}{s}}\);
  • Critical Values: \(t_{1-\alpha/2, n-1}\) (which equals \(\boxed{- t_{\alpha/2, n-1}}\)) and \(\boxed{t_{\alpha/2, n-1}}\);
  • Rejection Rule: Reject \(H_0\) if \(T > t_{\alpha/2, n-1}\) or \(T < - t_{\alpha/2, n-1}\), which is equivalent to \(\boxed{|T| > t_{\alpha/2, n-1}}\).

• Let \(X\) be the score of a student, then we know
  • \(X\) follows a Normal distribution;
  • the population mean of \(X\), denoted by \(\mu\), is unknown;
  • the population standard deviation of \(X\), denoted by \(\sigma\), is unknown;
  • the sample size \(n = 16\);
  • the sample mean \(\bar{X} = 79\);
  • the sample standard deviation \(s = 3.5\).
• To construct the confidence interval of \(\mu\) when \(\sigma\) is unknown,
  • the confidence level is \(1 - \alpha = 90\%\), so \(\alpha = 0.1\);
  • the midpoint of the confidence interval is \(\bar{X} = 79\);
  • the critical value is \(t_{\alpha/2, n-1} = t_{0.1/2, 16-1} = t_{0.05, 15} = 1.753\);
  • the margin of error is \(t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}} = 1.753 \cdot \frac{3.5}{\sqrt{16}} = 1.534\);
  • the lower confidence limit is \(79 - 1.534 = 77.466\);
  • the upper confidence limit is \(79 + 1.534 = 80.534\);
  • the confidence interval is \([77.466, 80.534]\);
  • the width of the confidence interval is \(2 \times 1.534 = 3.068\).

• Here we want to confirm the claim that \(\mu < 80\), so we let it be the alternative hypothesis. That is, we want to test \(H_0: \mu \ge 80\) vs \(H_A: \mu < 80\) (or equivalently \(H_0: \mu = 80\) vs \(H_A: \mu < 80\));
• The test statistic is \(T = \frac{\sqrt{n}(\bar{X} - \mu_0)}{s} = \frac{\sqrt{16}(79 - 80)}{3.5} = -1.143\);
• The significance level is \(\alpha = 5\%\);
• The critical value is \(- t_{\alpha, n-1} = - t_{0.05, 16-1} = - 1.753\);
• The rejection rule is: reject \(H_0\) if \(T < - t_{\alpha, n-1}\);
• Here \(T = -1.143 > -1.753 = -t_{\alpha, n-1}\), so we don't have enough evidence to reject \(H_0\) and thus don't have enough evidence to say that the population mean is lower than \(80\).

• The hypotheses are the same, \(H_0: \mu \ge 80\) vs \(H_A: \mu < 80\) (or equivalently \(H_0: \mu = 80\) vs \(H_A: \mu < 80\));
• The test statistic is the same, \(T = \frac{\sqrt{n}(\bar{X} - \mu_0)}{s} = \frac{\sqrt{16}(79 - 80)}{3.5} = -1.143\);
• The significance level now is \(\alpha = 10\%\);
• The critical value now is \(-t_{\alpha, n-1} = -t_{0.1, 16-1} = - 1.341\);
• The rejection rule is the same: reject \(H_0\) if \(T < - t_{\alpha, n-1}\);
• Here \(T = -1.143 > -1.341 = -t_{\alpha, n-1}\), so we still don't have enough evidence to reject \(H_0\) and thus still don't have enough evidence to say that the population mean is lower than \(80\).

• Midpoint: \(\boxed{ \hat{p} }\);
• Critical Value: \(\boxed{ Z_{\alpha/2} }\);
• Margin of Error: \(\boxed{ Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} }\);
• Lower Confidence Limit (LCL): \(\boxed{ \hat{p} - Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} }\);
• Upper Confidence Limit (UCL): \(\boxed{ \hat{p} + Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} }\);
• Confidence Interval: \(\boxed{ \left[ \hat{p} - Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hat{p} + Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \right] }\);
• Width of Confidence Interval: \(\boxed{ 2 \cdot Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} }\);

At confidence level \(1-\alpha\), to achieve a confidence interval for the population proportion \(p\) whose margin of error is no wider than \(M\), the smallest sample size needed is \[ \boxed{ n = \left( \frac{Z_{\alpha/2}}{2M} \right)^2 } \]

If the value of \(n\) calculated by the formula is not an integer, then we should round it up in order to ensure the actual margin of error is no wider than \(M\).