Tests from confidence intervals . A confidence interval for the population mean

Question

Tests from confidence intervals. A confidence interval for the population mean ? tells us which values of ? are plausible (those inside the interval) and which values are not plausible (those outside the interval) at the chosen level of confidence. You can use this idea to carry out a test of any null hypothesis H0: ? = ?0 starting with a confidence interval: reject H0 if ?0 is outside the interval and fail to reject if ?0 is inside the interval.

(b)The hypothesized value ?0 = 128 falls inside this confidence interval. Carry out the z test for H0: ? = 128 against the two-sided alternative. Show that the test is not statistically significant at the 10% level.

(c)The hypothesized value ?0 = 129 falls outside this confidence interval. Carry out the z test for H0: ? = 129 against the two-sided alternative. Show that the test is statistically significant at the 10% level.

Explanation / Answer

a)

CI for 90%

n = 72

mean = 126.07

z-value of 90% CI = 1.6449

std. dev. = 15

SE = std.dev./sqrt(n) = 1.76777

ME = z*SE = 2.90772

Lower Limit = Mean - ME = 123.16228

Upper Limit = Mean + ME = 128.97772

90% CI (123.1623 , 128.9777 )

b)

Test statistics,

z = (126.07 - 128)/(15/sqrt(72)) = -1.0918

p-value = 0.2749

As p-value is greater than 0.1, we fail to reject the null hypothesis.

this means test is not significant

c)

Test statistics,

z = (126.07 - 129)/(15/sqrt(72)) = -1.6575

p-value = 0.0974

As p-value is less than 0.1, we reject the null hypothesis.

this means test is significant

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