Confidence interval for mean of a normal population
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Consider a normal population with unknown mean $mu$ and variance $sigma^2=9$. To test $H_{0}:mu=0$,against $H_{1}:mu ne 0$. A random sample of size 100 is taken. Based on this sample, the test of the form $|bar{X_n}| > K$ rejects the null hypothesis at 5% level of significance. Then, which of the following is a possible 95% confidence interval for $mu$ ?
(A) (-0.488, 0.688) (B) (-1.96, 1.96)
(C) (0.422, 1.598) (D) (0.588, 1.96)
Now, if we construct the test statistic $tau=frac{bar{X_n}-mu}{0.3} sim N(0,1)$
Therefore our test ,actually based on the observed value of $tau=tau_{0}$ is given by $|tau_{0}|> tau_{frac{alpha}{2}}$ where $alpha=0.05$
Now,we get $K=frac{3}{10}tau_{0.025}=0.588$
Thus the 95% confidence interval for $mu$ is coming to be $(bar{X_n}-0.588 ,bar{X_n}+0.588)$So, looking at the options it seems that both (A) , (C) are correct but only (C) is given to be correct.
Please help!
probability probability-theory statistics statistical-inference confidence-interval
asked Dec 2 at 14:46
Legend Killer
1,556523
add a comment |
up vote
2
down vote
favorite
Consider a normal population with unknown mean $mu$ and variance $sigma^2=9$. To test $H_{0}:mu=0$,against $H_{1}:mu ne 0$. A random sample of size 100 is taken. Based on this sample, the test of the form $|bar{X_n}| > K$ rejects the null hypothesis at 5% level of significance. Then, which of the following is a possible 95% confidence interval for $mu$ ?
(A) (-0.488, 0.688) (B) (-1.96, 1.96)
(C) (0.422, 1.598) (D) (0.588, 1.96)
Now, if we construct the test statistic $tau=frac{bar{X_n}-mu}{0.3} sim N(0,1)$
Therefore our test ,actually based on the observed value of $tau=tau_{0}$ is given by $|tau_{0}|> tau_{frac{alpha}{2}}$ where $alpha=0.05$
Now,we get $K=frac{3}{10}tau_{0.025}=0.588$
Thus the 95% confidence interval for $mu$ is coming to be $(bar{X_n}-0.588 ,bar{X_n}+0.588)$So, looking at the options it seems that both (A) , (C) are correct but only (C) is given to be correct.
Please help!
probability probability-theory statistics statistical-inference confidence-interval
asked Dec 2 at 14:46
Legend Killer
1,556523
Consider the width of the CI.
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:51
sorry,i edited one of the options
– Legend Killer
Dec 2 at 14:52
@GNUSupporter8964民主女神地下教會they have the same width,I mean options A,C
– Legend Killer
Dec 2 at 14:55
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Consider a normal population with unknown mean $mu$ and variance $sigma^2=9$. To test $H_{0}:mu=0$,against $H_{1}:mu ne 0$. A random sample of size 100 is taken. Based on this sample, the test of the form $|bar{X_n}| > K$ rejects the null hypothesis at 5% level of significance. Then, which of the following is a possible 95% confidence interval for $mu$ ?
(A) (-0.488, 0.688) (B) (-1.96, 1.96)
(C) (0.422, 1.598) (D) (0.588, 1.96)
Now, if we construct the test statistic $tau=frac{bar{X_n}-mu}{0.3} sim N(0,1)$
Therefore our test ,actually based on the observed value of $tau=tau_{0}$ is given by $|tau_{0}|> tau_{frac{alpha}{2}}$ where $alpha=0.05$
Now,we get $K=frac{3}{10}tau_{0.025}=0.588$
Thus the 95% confidence interval for $mu$ is coming to be $(bar{X_n}-0.588 ,bar{X_n}+0.588)$So, looking at the options it seems that both (A) , (C) are correct but only (C) is given to be correct.
Please help!
probability probability-theory statistics statistical-inference confidence-interval
asked Dec 2 at 14:46
Legend Killer
1,556523
Consider a normal population with unknown mean $mu$ and variance $sigma^2=9$. To test $H_{0}:mu=0$,against $H_{1}:mu ne 0$. A random sample of size 100 is taken. Based on this sample, the test of the form $|bar{X_n}| > K$ rejects the null hypothesis at 5% level of significance. Then, which of the following is a possible 95% confidence interval for $mu$ ?
(A) (-0.488, 0.688) (B) (-1.96, 1.96)
(C) (0.422, 1.598) (D) (0.588, 1.96)
Now, if we construct the test statistic $tau=frac{bar{X_n}-mu}{0.3} sim N(0,1)$
Therefore our test ,actually based on the observed value of $tau=tau_{0}$ is given by $|tau_{0}|> tau_{frac{alpha}{2}}$ where $alpha=0.05$
Now,we get $K=frac{3}{10}tau_{0.025}=0.588$
Thus the 95% confidence interval for $mu$ is coming to be $(bar{X_n}-0.588 ,bar{X_n}+0.588)$So, looking at the options it seems that both (A) , (C) are correct but only (C) is given to be correct.
Please help!
probability probability-theory statistics statistical-inference confidence-interval
probability probability-theory statistics statistical-inference confidence-interval
asked Dec 2 at 14:46
Legend Killer
1,556523
asked Dec 2 at 14:46
Legend Killer
1,556523
edited Dec 2 at 14:52
asked Dec 2 at 14:46
Legend Killer
1,556523
asked Dec 2 at 14:46
Legend Killer
1,556523
asked Dec 2 at 14:46
Legend Killer
1,556523
1,556523
Consider the width of the CI.
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:51
sorry,i edited one of the options
– Legend Killer
Dec 2 at 14:52
@GNUSupporter8964民主女神地下教會they have the same width,I mean options A,C
– Legend Killer
Dec 2 at 14:55
add a comment |
Consider the width of the CI.
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:51
sorry,i edited one of the options
– Legend Killer
Dec 2 at 14:52
@GNUSupporter8964民主女神地下教會they have the same width,I mean options A,C
– Legend Killer
Dec 2 at 14:55
Consider the width of the CI.
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:51
Consider the width of the CI.
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:51
sorry,i edited one of the options
– Legend Killer
Dec 2 at 14:52
sorry,i edited one of the options
– Legend Killer
Dec 2 at 14:52
@GNUSupporter8964民主女神地下教會they have the same width,I mean options A,C
– Legend Killer
Dec 2 at 14:55
@GNUSupporter8964民主女神地下教會they have the same width,I mean options A,C
– Legend Killer
Dec 2 at 14:55
add a comment |
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Consider the width of the CI.
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:51
sorry,i edited one of the options
– Legend Killer
Dec 2 at 14:52
@GNUSupporter8964民主女神地下教會they have the same width,I mean options A,C
– Legend Killer
Dec 2 at 14:55