The researcher claim is invalid and the p-value based on alpha = 0.05 is approx. = 0.
In the question ,
it is given that ,
a researches claims that per person yearly consumption (μ) is 52 gallons .
sample size (n) = 50 .
the mean of the consumption (x) = 56.3 .
the standard deviation (σ) is = 3.5 gallons .
the α = 0.05 .
let the null hypothesis ,
H₀ : μ = 52 and
H₁ : μ ≠ 52 .
the test statistics is : z = (x - μ)/(σ/√n)
the critical value ( rejection region ) : {z : |z| ≥ z₀.₀₂₅ = 1.96 }
So , z = (56.3 - 52)/(3.5/√50)
Simplifying further ,
we get ,
z = 8.69
We reject the null hypothesis H₀ , since the absolute value of the test statistics 8.69 is greater than the critical value 1.96 .
Since this is a two tailed test, p-value must be multiplied by two
2 × P(z > 8.69) ≈ 0
Since the p-value is less than α(0.05), So ,we reject the claim of the researcher that the mean yearly consumption of soft drinks per person is 52 gallons
Hence , the claim is not valid .
Therefore , The claim is invalid and p-value is ≈ 0 .
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