CHI-squared Test Example for df = 4 at α = 0.1
Solved example work with steps, formula and summary to estimate χ²-statistic (χ²0), critical (table) value (χ²e) for given degrees of freedom and hypothesis test (H0) at a stated level of significance (α = 0.1) for observed frequencies O = {15, 45, 105, . . . . , 9} and expected frequencies E = {68, 90, 206, . . . . , 100} to check if the result of this probability and statistics experiment is statistically significant.| Calculation Summary | |
|---|---|
| Observed Frequency (O) | {15, 45, 105, . . . . , 9} |
| Expected Frequency (E) | {68, 90, 206, . . . . , 100} |
| Significance Level (α) | 0.1 |
| χ²0 | 376.8441 |
| χ²e | 7.779 |
Workout for O = {15, 45, 105, . . . . , 9} and E = {68, 90, 206, . . . . , 100}
The below is the solved example with step by step calculation shows how to estimate the chi-squared statistic (χ²0), critical (table) value (χ²e) for degrees of freedom and hypothesis test (H0) at a stated level of significance (α = 0.1) for observed frequencies O = {15, 45, 105, . . . . , 9} and expected frequencies E = {68, 90, 206, . . . . , 100} to check the quality of variances between two or more samples which are not uniformly distributed, may help students and learners solve similar chi-squared test (χ²0, χ²e and H0) worksheet problems efficiently.
Workout :
step 1 Address the formula input parameters and values
observed frequency = 15, 45, 105, . . . . , 9
expected frequency = 68, 90, 206, . . . . , 100
Significance Level (α) = 0.1
step 2 Refer the below χ² table to find χ²-statistic (χ²0)
χ² = ∑(Oi - Ei)2Ei
χ²0 = 376.8441
step 3 Find the degrees of freedom
df = n - 1
df = 5 - 1
df = 4
step 4 Find the critical value of χ²e for df = 4 at a stated level of significance α = 0.1 from χ²-distribution table.
χ²e = 7.779
Inference
for χ²0 > χ²e
There is significance difference between sample and population
since χ²0 = 376.8441 is greater than χ²e = 7.779.
Therefore the null hypothesis H0 is rejected.
step 1 Address the formula input parameters and values
observed frequency = 15, 45, 105, . . . . , 9
expected frequency = 68, 90, 206, . . . . , 100
Significance Level (α) = 0.1
step 2 Refer the below χ² table to find χ²-statistic (χ²0)
χ² = ∑(Oi - Ei)2Ei
| Observed Frequency (O) | Expected Frequency (E) | (O - E)2 | (O - E)2E |
| 15 | 68 | 2809 | 41.3088 |
| 45 | 90 | 2025 | 22.5 |
| 105 | 206 | 10201 | 49.5194 |
| 12 | 204 | 36864 | 180.7059 |
| 9 | 100 | 8281 | 82.81 |
| ∑(O - E)2E = 376.8441 | |||
step 3 Find the degrees of freedom
df = n - 1
df = 5 - 1
df = 4
step 4 Find the critical value of χ²e for df = 4 at a stated level of significance α = 0.1 from χ²-distribution table.
χ²e = 7.779
Inference
for χ²0 > χ²e
There is significance difference between sample and population
since χ²0 = 376.8441 is greater than χ²e = 7.779.
Therefore the null hypothesis H0 is rejected.
