a)π1,β¦,ππβΌπ.π.π.π―ππππ(π) for some unknown π>0 ;
π»0:π=π0 v.s. π»1:πβ π0where π0>0.
(Type barX_n for πβ―β―β―β―β―π , lambda_0 for π0 . If applicable, type abs(x) for |π₯| , Phi(x) for Ξ¦(π₯)=π(πβ€π₯) where πβΌN(0,1) , and q(alpha) for ππΌ , the 1βπΌ quantile of a standard normal variable, e.g. enter q(0.01) for π0.01 .)
Asymptotic π-value=?
b) π1,β¦,ππβΌπ.π.π.π―ππππ(π) for some unknown π>0 ;
π»0:πβ₯π0 v.s. π»1:π<π0where π0>0.
( Type barX_n for πβ―β―β―β―β―π , lambda_0 for π0. . If applicable, type abs(x) for |π₯| , Phi(x) for Ξ¦(π₯)=π(πβ€π₯) where πβΌN(0,1) , and q(alpha) for ππΌ , the 1βπΌ quantile of a standard normal variable. )
Asymptotic π-value=?
c)π1,β¦,ππβΌπ.π.π.π€ππ(π) for some unknown π>0 ;
π»0:π=π0 v.s. π»1:πβ π0where π0>0.
(Type barX_n for πβ―β―β―β―β―π , lambda_0 for π0. If applicable, type abs(x) for |π₯| , Phi(x) for Ξ¦(π₯)=π(πβ€π₯) where πβΌN(0,1) , and q(alpha) for ππΌ , the 1βπΌ quantile of a standard normal variable.)
Asymptotic π-value=?
a) 2*Phi((sqrt(n)*(barX_n-lambda_0))/sqrt(lambda_0))
b) 1-Phi((sqrt(n)*(lambda_0-barX_n))/sqrt(lambda_0))
c) 2*(1-Phi(sqrt(n)*(barX_n-(1/lambda_0))*lambda_0))
a) To calculate the asymptotic π-value for the test π»0:π=π0 v.s. π»1:πβ π0, where π1,β¦,ππ are π.π.π. ππππ π (π) random variables with π>0, you can use the formula:
π-value = 2 * (1 - Phi(sqrt(n) * abs(barX_n - lambda_0) / sqrt(lambda_0)))
where barX_n is the sample mean, lambda_0 is the assumed value of π, abs(x) is the absolute value of x, Phi(x) is the cumulative distribution function of the standard normal distribution, and n is the sample size.
b) To calculate the asymptotic π-value for the test π»0:πβ₯π0 v.s. π»1:π<π0, where π1,β¦,ππ are π.π.π. ππππ π (π) random variables with π>0, you can use the formula:
π-value = Phi(sqrt(n) * (barX_n - lambda_0) / sqrt(lambda_0))
where barX_n is the sample mean, lambda_0 is the assumed value of π, abs(x) is the absolute value of x, Phi(x) is the cumulative distribution function of the standard normal distribution, and n is the sample size.
c) To calculate the asymptotic π-value for the test π»0:π=π0 v.s. π»1:πβ π0, where π1,β¦,ππ are π.π.π. πΈπ₯π(π) random variables with π>0, you can use the formula:
π-value = 2 * (1 - Phi(sqrt(n) * abs(barX_n - lambda_0) / sqrt(lambda_0)))
where barX_n is the sample mean, lambda_0 is the assumed value of π, abs(x) is the absolute value of x, Phi(x) is the cumulative distribution function of the standard normal distribution, and n is the sample size.
a) To calculate the asymptotic p-value for testing π»0:π=π0 versus π»1:πβ π0, we can use the likelihood ratio test statistic.
The likelihood ratio test statistic is given by:
πΜ = -2 ln [(likelihood under π»0)/(likelihood under π»1)]
Under the null hypothesis π»0, the likelihood function is πΏ(π0) = π0^π * exp(-π0 * πβ―β―β―β―β―π), where πβ―β―β―β―β―π is the sample mean.
Under the alternative hypothesis π»1, the likelihood function is πΏ(πΜ) = πΜ^π * exp(-πΜ * πβ―β―β―β―β―π).
To calculate the p-value, we need to find the probability of observing a test statistic as extreme as πΜ under the null hypothesis.
The asymptotic p-value can be calculated using the chi-squared distribution with 1 degree of freedom. Since we have 1 parameter estimated under the null hypothesis, the test statistic follows a chi-squared distribution with 1 degree of freedom under the null hypothesis.
Therefore, the asymptotic p-value is given by:
π-value = π(πΜ > π* | π»0) = 1 - Ξ¦(sqrt(πΜ))
where Ξ¦(x) is the cumulative distribution function (CDF) of the standard normal distribution and π* is the limit of the chi-squared distribution with 1 degree of freedom.
b) To calculate the asymptotic p-value for testing π»0:πβ₯π0 versus π»1:π<π0, we can follow a similar approach as in part a).
The likelihood ratio test statistic is still given by:
πΜ = -2 ln [(likelihood under π»0)/(likelihood under π»1)]
Under the null hypothesis π»0, the likelihood function is πΏ(π0) = π0^π * exp(-π0 * πβ―β―β―β―β―π), where πβ―β―β―β―β―π is the sample mean.
Under the alternative hypothesis π»1, the likelihood function is πΏ(πΜ) = πΜ^π * exp(-πΜ * πβ―β―β―β―β―π).
To calculate the p-value, we need to find the probability of observing a test statistic as extreme as πΜ under the null hypothesis.
The asymptotic p-value can be calculated using the chi-squared distribution with 1 degree of freedom. Since we have 1 parameter estimated under the null hypothesis, the test statistic follows a chi-squared distribution with 1 degree of freedom under the null hypothesis.
Therefore, the asymptotic p-value is given by:
π-value = π(πΜ > π* | π»0) = π(πΜ > π* | π»0:π=π0) = 1 - Ξ¦(sqrt(πΜ))
where Ξ¦(x) is the cumulative distribution function (CDF) of the standard normal distribution and π* is the limit of the chi-squared distribution with 1 degree of freedom.
c) To calculate the asymptotic p-value for testing π»0:π=π0 versus π»1:πβ π0 for exponential distribution, we can use a similar approach as in part a).
The likelihood ratio test statistic is still given by:
πΜ = -2 ln [(likelihood under π»0)/(likelihood under π»1)]
Under the null hypothesis π»0, the likelihood function is πΏ(π0) = π0^π * exp(-π0 * πβ―β―β―β―β―π), where πβ―β―β―β―β―π is the sample mean.
Under the alternative hypothesis π»1, the likelihood function is πΏ(πΜ) = πΜ^π * exp(-πΜ * πβ―β―β―β―β―π).
To calculate the p-value, we need to find the probability of observing a test statistic as extreme as πΜ under the null hypothesis.
The asymptotic p-value can be calculated using the chi-squared distribution with 1 degree of freedom. Since we have 1 parameter estimated under the null hypothesis, the test statistic follows a chi-squared distribution with 1 degree of freedom under the null hypothesis.
Therefore, the asymptotic p-value is given by:
π-value = π(πΜ > π* | π»0) = 1 - Ξ¦(sqrt(πΜ))
where Ξ¦(x) is the cumulative distribution function (CDF) of the standard normal distribution and π* is the limit of the chi-squared distribution with 1 degree of freedom.
To calculate the asymptotic p-value for these hypothesis tests, we can use the likelihood ratio test statistic.
a) For the hypothesis test π»0:π=π0 v.s. π»1:πβ π0, where π0>0, we can calculate the test statistic as follows:
1. Compute the sample mean: barX_n = (π1 + π2 + ... + ππ)/π.
2. Calculate the likelihood ratio:
LR = 2 * (log(likelihood(π0)) - log(likelihood(barX_n))).
The log-likelihood function for a Poisson distribution is given by log(likelihood(π)) = π * (π * log(π) - π - log(π1!) - log(π2!) - ... - log(ππ!)).
3. Calculate the p-value using the chi-square distribution:
p-value = 1 - Ξ¦(sqrt(LR)).
b) For the hypothesis test π»0:πβ₯π0 v.s. π»1:π<π0, where π0>0, we can calculate the test statistic as follows:
1. Compute the sample mean: barX_n = (π1 + π2 + ... + ππ)/π.
2. Calculate the likelihood ratio:
LR = 2 * (log(likelihood(π_0)) - log(likelihood(barX_n))).
The log-likelihood function for a Poisson distribution is given by log(likelihood(π)) = π * (π * log(π) - π - log(π1!) - log(π2!) - ... - log(ππ!)).
3. Calculate the p-value using the chi-square distribution:
p-value = Ξ¦(sqrt(LR)).
c) For the hypothesis test π»0:π=π0 v.s. π»1:πβ π0, where π0>0, we can calculate the test statistic as follows:
1. Compute the sample mean: barX_n = (π1 + π2 + ... + ππ)/π.
2. Calculate the likelihood ratio:
LR = 2 * (log(likelihood(π0)) - log(likelihood(barX_n))).
The log-likelihood function for an exponential distribution is given by log(likelihood(π)) = π * (log(π) - π * π1 - π * π2 - ... - π * ππ).
3. Calculate the p-value using the chi-square distribution:
p-value = 1 - Ξ¦(sqrt(LR)).
Note: In all three cases, Ξ¦(x) represents the cumulative distribution function of the standard normal distribution, and q(alpha) represents the (1βalpha) quantile of the standard normal distribution.