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Degrees of freedom and confidence interval calculations

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1. The P-value for a two sided test of the null hypothesis H0: µ = 30 is 0.04.
a) Does the 95% confidence interval include the value 30? Why?
b) Does the 90% confidence interval include the value 30? Why?

2. The level of calcium in the blood in healthy young adults varies with mean about 9.5 milligrams per deciliter and standard deviation about σ= 0.4. A clinic in rural Guatemala measures the blood calcium level of 160 healthy pregnant women at their first visit for prenatal care. The mean is = 9.57. Is this an indication that the mean calcium level in the population from which these women come differs from 9.5?

a) State H0 and Ha.
b) Carry out the test and give the P-value, assuming that σ= .4 in this population. Report your conclusion.
c). Give a 95% confidence interval for the mean calcium level µ in this population. We are confident that µ lies quite close to 9.5. This illustrates the fact that a test based on a large sample (n=160 here) will often declare even a small deviation from H0 to be statistically significant.

3. The one-sample t statistic for testing
H0: µ=10
Ha: µ > 10
from a sample of n=20 observations has the value t=2.10.

a) What are the degrees of freedom for this statistic?
b) Give the two critical values t* from Table D that bracket t.
c) Between what two values does the P-value of the test fall?
d) Is the value t=2.10 significant at the 5% level? Is it significant at the 1% level?
e) If you have software available, find the exact P-value?

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1. The P-value for a two sided test of the null hypothesis H0: µ = 30 is 0.04.
a) Does the 95% confidence interval include the value 30? Why?
No. A 95% confidence interval corresponds to a hypothesis test with significance level of .05. A 95% confidence interval contains the hypothesized value only if the p-value is greater than .05. If the p-value is less than the .05 level of significance, the confidence interval does not contain the hypothesized value.
b) Does the 90% confidence interval include the value 30? Why?
No. A 90% confidence interval corresponds to a hypothesis test with significance level of .10. A 90% confidence interval contains the hypothesized value only if the p-value is greater than .10. If the p-value is less than the .10 level of significance, the confidence interval does not contain the hypothesized value.
2. The level of calcium in the blood in healthy young adults varies with mean about 9.5 milligrams per deciliter and standard deviation about σ= 0.4. A clinic in rural Guatemala measures the blood calcium level of 160 healthy pregnant women ...

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