Question

Find the critical value needed to construct a confidence interval of the given level with the given sample size. Round the answers to three decimal places.
Part 1 of 4
(a) For level 95% and sample size 7.
Part 2 of 4
(b) For level 98% and sample size 12.
Part 3 of 4
(c) For level 99% and sample size 27.
Part 4 of 4
(d) For level 98% and sample size 14.

Answer 1

Answer:

a) $t_{\alpha/2}=\pm 2.447$

b) $t_{\alpha/2}=\pm 2.718$

c) $t_{\alpha/2}=\pm 2.779$

d) $t_{\alpha/2}=\pm 2.650$

Step-by-step explanation:

Part a

The degrees of freedom are given by:

$df=n-1= 7-1=6$

The confidence is 95% or 0.95 the significance would be $\alpha=0.05$ and the critical values would be:

$t_{\alpha/2}=\pm 2.447$

Part b

The degrees of freedom are given by:

$df=n-1=12-1=11$

The confidence is 98% or 0.98 the significance would be $\alpha=0.02$ and the critical values would be:

$t_{\alpha/2}=\pm 2.718$

Part c

The degrees of freedom are given by:

$df=n-1=27-1=26$

The confidence is 99% or 0.99 the significance would be $\alpha=0.01$ and the critical values would be:

$t_{\alpha/2}=\pm 2.779$

Part d

The degrees of freedom are given by:

$df=n-1=14-1=13$

The confidence is 98% or 0.98 the significance would be $\alpha=0.02$ and the critical values would be:

$t_{\alpha/2}=\pm 2.650$

Answer 2

A value of a test statistic defines the upper bounds of a confidence interval, statistical significance in a test statistic is known as the crucial value.

critical value:

For part a )

n = 7

Calculating the degrees of freedom $= df = n - 1 = 7 - 1 = 6$

At $95\%$ the confidence level, the t:

$\alpha = 1 - 95\% = 1 - 0.95 = 0.05\\\\\frac{\alpha}{ 2} = \frac{0.05}{ 2} = 0.025\\\\t_{\frac{\alpha}{ 2}}\\\\df = t_{0.025,6} = 2.447$

The critical value = 2.447

For part b )

n =12

Calculating the degrees of freedom$= df = n - 1 = 12 - 1 = 11$

At $98\%$ the confidence level , the t:

$\alpha = 1 - 98\% = 1 - 0.98 = 0.02\\\\\frac{\alpha}{ 2}= \frac{0.02}{ 2} = 0.01\\\\t_{\frac{\alpha}{ 2}}\\\\df = t_{0.01,11} =2.718$

The critical value =2.718

For part c )

n = 27

Calculating the degrees of freedom $= df = n - 1 = 27 - 1 = 26$

At $99\%$ the confidence level, the t:

$\alpha = 1 - 99\% = 1 - 0.99 = 0.01\\\\\frac{\alpha}{ 2} = \frac{0.01}{ 2} = 0.005\\\\t_{\frac{\alpha}{ 2}}\\\\df = t_{0.005,26}=2.779$

The critical value =2.779

For part d )

n = 14

Calculating the degrees of freedom $= df = n - 1 = 14 - 1 = 13$

At $98\%$ the confidence level, the t:

$\alpha = 1 - 98\% = 1 - 0.98 = 0.02\\\\\frac{\alpha}{ 2} = \frac{0.02}{ 2} = 0.01\\\\t_{\frac{\alpha}{ 2}} \\\\df = t_{0.01,13} =2.650$

The critical value =2.650

Find out more about the critical value here:

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