List of top Statistics Questions on Testing of Hypotheses asked in IIT JAM MS

The observed value of a random sample of size \(9\) from a distribution having continuous and strictly increasing cumulative distribution function is as below:

Let \(X_1,X_2,X_3,X_4\) be a random sample of size \(4\) from a \(\chi_m^2\) distribution, where \(m\in \mathbb{N}\) is an unknown parameter. To test \(H_0:m=1\) against \(H_1:m=2\), the critical region \(\sum_{i=1}^{4}X_i>6\) is being used. If \(\alpha\) and \(\beta\) denote the probabilities of Type-I error and Type-II error, respectively, then which one of the following statements is true?

Let \(X\) be a random sample of size \(1\) from a \(U(0,\theta)\) distribution, where \(\theta>0\) is an unknown parameter. To test \(H_0:\theta\geq 1\) against \(H_1:\theta<1\), the critical region \(X<\frac{3}{4}\) is being used. If \(\beta(\cdot)\) is the power function of the test, then which one of the following statements is NOT true?

The recorded air quality index (AQI) at a place in a city for \(16\) consecutive days for the current year is compared with the historical average AQI. The direction of the recorded AQI is noted as \(+\) if it is above the historical average and as \(-\) if it is below the historical average. The dataset is as below:

Let \(X_1,X_2,\ldots,X_n\) be a random sample of size \(n\ (n\geq2)\) from a \(N(\mu,1)\) distribution, where \(\mu\in\mathbb{R}\) is an unknown parameter. Consider the problem of testing \(H_0:\mu=5\) against \(H_1:\mu\neq5\). Which one of the following statements is true?