The hypothesis, at the 0.05 significance level,
H₀ : the observed distribution of X may be fitted by the geometric distribution
Hₐ : the observed distribution of X may not be fitted by the geometric distribution
we conclude p-Value > α so, cannot reject null hypothesis. The observed distribution of X may be fitted by the geometric distribution g(x ; 1 / 2), x=1,2,3
A coin is thrown until a head occurs and the number X of tosses recorded.
The probability that head occur on toss ,p = 0.5
Consider the null and alternative hypothesis,
H₀ : the observed distribution of X may be fitted by the geometric distribution
Hₐ : the observed distribution of X may not be fitted by the geometric distribution
The chi square test statistic here as:
χ² = ∑( Oᵢ - Eᵢ)²/Eᵢ
where Oᵢ--> observed value
Eᵢ --> expected value
and the expected frequencies for each X here E(x) = P(X = x) = 256×g(x)
The probability mass function of geometric distribution with parameter 1/2 is g(x) = (1/2)(1 - 1/2)⁽ˣ⁻¹⁾
where x = 0, 1,2 ,3,......
Observed values are f :136 60 34 12 9 1 3 1
Using the data,we calculate the expected values
E₁ = P( x = 1) = 256×g(1) = 256×(1/2)(1/2)⁰
= 128
E₂ = P( x =2) = 256×g(2) = 256×(1/2)(1/2)¹
= 64
E₃ = P( x = 3) = 256×g(3) = 256×(1/2)(1/2)² = 32
E₄ = P( x = 4) = 256×g(4) = 256×(1/2)(1/2)³
= 16
E₅ = P( x = 5) = 256×g(5) = 256×(1/2)(1/2)⁴
= 8
E₆ = P( x = 6) = 256×g(6) = 256×(1/2)(1/2)⁵
= 4
E₇ = P( x = 7) = 256×g(7) = 256×(1/2)(1/2)⁶
= 2
E₈ = P( x = 8) = 256×g(8) = 256×(1/2)(1/2)⁷
= 1
Hence, χ² = ∑( Oᵢ - Eᵢ)²/Eᵢ
= (136 - 128)²/128 + (60 - 64)²/64 + (34 - 32)²/32 + (12-16)²/16 +(9 - 8)²/8 + (1 - 4 )²/4 + (3 - 2)²/2 +(1 - 1)²/1
= 0.5 + 0.25 + 0.125 + 1 + 0.125 + 2.25 + 0.5 =4.75
Significance level is α =0.05
Using the χ² table, the p- value for χ² is 0.6290 .. Now P value > 0.05 , therefore the test is not significant here and we cannot reject the null hypothesis here. Therefore we have sufficient evidence here that the frequencies given are fitted by the geometric (p = 0.5) distribution.
To learn more about Chi-Square test, refer:
https://brainly.com/question/4543358
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