ICAR AIEEA PG Statistical Sciences 09 July 2023 (Shift 1) Paper

Analysis of critical regions in statistical tests:

• Correct: When the alternative hypothesis specifies a value greater than the hypothesized parameter ($H_1: \phi \gt 5$), the rejection region lies entirely in the upper (right) tail of the test statistic's sampling distribution, making it a right-tailed test.
• Incorrect: A left-tailed test corresponds to $H_1: \phi \lt 5$.
• Incorrect: 'Center tailed' is non-standard; tests with $H_1: \phi \ne 5$ are two-tailed.
• Incorrect: 'Cross tailed' is not a statistical concept.

Mathematical derivation of original score from Z-score:
Given $Z = 2.5$, $\mu = 45$, and $\sigma = 3$.

• Incorrect: 42 is obtained from $45 - 3$, which is $\mu - \sigma$.
• Correct: Using the definition of standard normal variate:
  $$Z = \frac{X - \mu}{\sigma}$$
  $$2.5 = \frac{X - 45}{3}$$
  $$X - 45 = 2.5 \times 3 = 7.5$$
  $$X = 45 + 7.5 = 52.5$$
• Incorrect: 47 is obtained if $Z = 2/3$.
• Incorrect: 3 is the standard deviation.

Step-by-step correction of mean and variance:

• Given $n = 200$, incorrect mean $\bar{X} = 65.7$, incorrect $\sigma = 4.4$.
  $$\sum X_{\text{incorrect}} = 200 \times 65.7 = 13140$$
• Correcting the sum of observations:
  $$\sum X_{\text{correct}} = 13140 - (91 + 80) + (71 + 83) = 13140 - 171 + 154 = 13123$$
  $$\bar{X}_{\text{correct}} = \frac{13123}{200} = 65.615 \approx 65.5$$
• Correcting sum of squares:
  $$\sigma^2 = \frac{\sum X^2}{n} - (\bar{X})^2 \implies 4.4^2 = 19.36 = \frac{\sum X^2}{200} - (65.7)^2$$
  $$\frac{\sum X^2}{200} = 19.36 + 4316.49 = 4335.85 \implies \sum X^2 = 867170$$
  $$\sum X^2_{\text{correct}} = 867170 - (91^2 + 80^2) + (71^2 + 83^2) = 867170 - 14681 + 11930 = 864419$$
  $$\sigma^2_{\text{correct}} = \frac{864419}{200} - (65.615)^2 = 4322.095 - 4305.328 = 16.767$$
  $$\sigma_{\text{correct}} = \sqrt{16.767} \approx 4.1 \approx 4.2$$
• Correct: Matches mean = 65.5 and s.d. = 4.2.
• Incorrect: Mean increases instead of decreasing.
• Incorrect: Variance unchanged is impossible.
• Incorrect: Mean of 65.0 is too low.