Calculators & Converters

    CHI-squared Test Example for df = 4 at α =

    Chi-squared (χ2) Test CalculatorSolved example work with steps, formula & summary to estimate χ²-statistic (χ²0), critical (table) value (χ²e) for given degrees of freedom & hypothesis test (H0) at a stated level of significance (α = ) for observed frequencies O = {15, 45, 105, . . . . , 9} & expected frequencies E = {68, 90, 206, . . . . , 100α=0.1} to check if the result of this probability & statistics experiment is statistically significant.
    Calculation Summary
    Observed Frequency (O){15, 45, 105, . . . . , 9}
    Expected Frequency (E){68, 90, 206, . . . . , 100α=0.1}
    Significance Level (α)
    χ²0376.8441
    χ²e0

    Workout for O = {15, 45, 105, . . . . , 9} & E = {68, 90, 206, . . . . , 100α=0.1}

    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 & hypothesis test (H0) at a stated level of significance (α = ) for observed frequencies O = {15, 45, 105, . . . . , 9} & expected frequencies E = {68, 90, 206, . . . . , 100α=0.1} to check the quality of variances between two or more samples which are not uniformly distributed, may help learners or grade school students to solve the similar chi-squared test (χ²0, χ²e & 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, . . . . , 100α=0.1
    Significance Level (α) =

    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
    1568280941.3088
    4590202522.5
    1052061020149.5194
    1220436864180.7059
    9100α=0.1828182.81
    (O - E)2E = 376.8441
    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 = from χ²-distribution table.
    χ²e = 0

    Inference
    for χ²0 > χ²e
    There is significance difference between sample and population
    since χ²0 = 376.8441 is greater than χ²e = 0.
    Therefore the null hypothesis H0 is rejected.

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