The problem asks for the $\chi^2$ (chi-square) value given the population variance, sample size, and the sum of squares of deviations from the sample mean.
Chi-Square Value Calculation
Given Information:
- Population variance: $\sigma^2 = 6$
- Sample size: $n = 15$
- Sum of squares of deviations from sample mean: $\sum (x_i - \bar{x})^2 = 120$
Step 1: Calculate Sample Variance ($s^2$)
The sample variance ($s^2$) is calculated using the sum of squares and the sample size ($n$). The formula is:
$ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} $
Substitute the given values:
$ s^2 = \frac{120}{15-1} = \frac{120}{14} $
Step 2: Calculate Chi-Square ($\chi^2$) Value
The formula for the chi-square statistic for testing population variance is:
$ \chi^2 = \frac{(n-1)s^2}{\sigma^2} $
Substitute the calculated sample variance and the given population variance:
$ \chi^2 = \frac{(15-1) \times \left(\frac{120}{14}\right)}{6} $
Simplify the expression:
$ \chi^2 = \frac{14 \times \left(\frac{120}{14}\right)}{6} $
$ \chi^2 = \frac{120}{6} $
$ \chi^2 = 20 $
Result
The calculated $\chi^2$ value is 20.
Important Questions from Hypothesis testing - Teaching
- The null hypothesis in nonparametric test often _______.
1. Includes specification of a population's parameters
2. Is used to evaluate some general population aspect
3. Is very similar to that used in regression analysis
4. Simultaneously tests more than two population parameters - The _______ test determines whether there is a significant difference between the observed and hypothesized distribution for a sample.
1. Independence
2. Coefficient of determination
3. Correlation analysis
4. Goodness-of-fit - Match the LIST-I with LIST-II
LIST-I LIST-II A. One-Tailed Test I. Null hypothesis is rejected if the sample value is significantly higher or lower than the hypothesized value of the population parameter B. Paired difference Test II. A hypothesis test of the difference between the sample means of two independent samples C. Two-Tailed Test III. A sample value significantly above the hypothesized population value will lead to rejection of the null hypothesis D. Upper-Tailed Test IV. Concerned only with whether the observed value deviates from the hypothesized value in one direction
Choose the correct answer from the options given below: - Arrange in correct sequence the steps for testing Statistical Hypothesis :
I. A random sample is drawn from population.
II. Specify the level of significance.
III. Value of the sample statistic is obtained.
IV. Obtained sample statistic is examined with the random sampling distribution of the statistic under consideration.
V. Formulate Null Hypothesis about a parameter of the population, along with an alternate hypothesis.
VI. Retain Null Hypothesis if particular sample outcome is in line with the expected outcomes, otherwise reject it and retain alternate hypothesis.
Codes : - Which set of stoichiometric coefficient correctly balance the equation ?
$$a. H_2O_2 + b. KMnO_4 + c. H_2SO_4 \to d. K_2SO_4 + e. MnSO_4 + f. H_2O + g. O_2$$
Select the correct answer from the codes given below :
