Use α \alpha to find the critical value(s), then determine if the given test statistic is in the rejection region. ( A ) \left(A\right) H 0 : μ = 7.5 ; H a : μ > 7.5 H_0:μ=7.5; H_a:μ>7.5 α = 0.01 ; z = 2.17 α=0.01; z=2.17 Test is [ LEFT | TWO | RIGHT ] -tailed Critical Value(s): Test stat [ IN | NOT IN ] rejection region.

Test is RIGHT -tailed, Critical value(s): 2.33 2.33 2.33 , Test stat NOT IN rejection region.

Test is TWO -tailed, Critical value(s): ± 2.33 \pm2.33 ± 2.33 , Test stat IN rejection region.

Test is LEFT -tailed, Critical value(s): − 2.33 -2.33 − 2.33 , Test stat NOT IN rejection region.

Test is RIGHT -tailed, Critical value(s): 2.17 2.17 2.17 , Test stat IN rejection region.

Use α \alpha to find the critical value(s), then determine if the given test statistic is in the rejection region. ( B ) \left(B\right) H 0 : p = 0.64 ; H a : p < 0.64 H_0:p=0.64; H_a:p<0.64 α = 0.10 ; z = − 1.53 α=0.10; z=-1.53 Test is [ LEFT | TWO | RIGHT ] -tailed Critical Value(s): Test stat [ IN | NOT IN ] rejection region.

Test is LEFT -tailed, Critical value(s): − 1.28 -1.28 − 1.28 , Test stat IN rejection region.

Test is LEFT -tailed, Critical value(s): − 1.28 -1.28 − 1.28 , Test stat NOT IN rejection region.

Test is TWO -tailed, Critical value(s): ± 1.64 \pm1.64 ± 1.64 Test stat IN rejection region.

Test is RIGHT -tailed, Critical value(s): 1.64 1.64 1.64 , Test stat IN rejection region.

Use α \alpha to find the critical value(s), then determine if the given test statistic is in the rejection region. ( C ) \left(C\right) H 0 : μ = 98.6 H_0:μ=98.6 ; H a : μ ≠ 98.6 H_a:μ≠98.6 α = 0.05 α=0.05 t = 2.41 ; d f = 17 t=2.41; df=17 Test is [ LEFT | TWO | RIGHT ] -tailed Critical Value(s): Test stat [ IN | NOT IN ] rejection region.

Test is TWO -tailed, Critical value(s): ± 2.11 \pm2.11 ± 2.11 , Test stat IN rejection region.

Test is RIGHT -tailed, Critical value(s): 2.11 2.11 2.11 , Test stat IN rejection region.

Test is RIGHT -tailed, Critical value(s): 2.11 2.11 2.11 , Test stat NOT IN rejection region.

Test is RIGHT -tailed, Critical value(s): 0.025 0.025 0.025 , Test stat IN rejection region.

Use α \alpha to find the critical value(s), then determine if the given test statistic is in the rejection region. ( D ) \left(D\right) H 0 : σ 2 = 0.3 ; H a : σ 2 > 0.3 H_0:σ^2=0.3; H_a:σ^2>0.3 α = 0.05 α=0.05 χ 2 = 61.7 ; d f = 51 χ^2=61.7; df=51 Test is [ LEFT | TWO | RIGHT ] -tailed Critical Value(s): Test stat [ IN | NOT IN ] rejection region.

Test is RIGHT -tailed, Critical value(s): 68.7 68.7 68.7 , Test stat NOT IN rejection region.

Test is RIGHT -tailed, Critical value(s): 68.7 68.7 68.7 , Test stat IN rejection region.

Test is RIGHT -tailed, Critical value(s): 61.7 61.7 61.7 , Test stat NOT IN rejection region.

Test is LEFT -tailed, Critical value(s): − 61.7 -61.7 − 61.7 , Test stat NOT IN rejection region.

A coffee shop owner believes the shop’s average daily sales are $1,200, with a known population standard deviation of $50. A manager claims it’s higher because of their new initiatives & collects a sample of 25 days and finds an average daily sales of $1,230. ( A ) \left(A\right) Use α = 0.10 α=0.10 & the critical value method to test if the true mean daily sales are higher than $1200. H 0 : μ = H_0: μ= _____ H a : μ H_a: μ ______ x ‾ = \overline{x}= ______ σ = \sigma= ______ n = n= ______ z = z= Because test stat. ( z ) (z) is [ INSIDE | OUTSIDE ] rejection region, we [ REJECT | FAIL TO REJECT ] H 0 H_0 . There is [ ENOUGH | NOT ENOUGH ] evidence to conclude that…

Because test stat. ( z ) (z) ( z ) is INSIDE rejection region, we REJECT H 0 H_0 H 0 . There is ENOUGH evidence to conclude that μ > 1200.

Because test stat. ( z ) (z) ( z ) is INSIDE rejection region, we REJECT H 0 H_0 H 0 . There is ENOUGH evidence to conclude that μ = 1200.

Because test stat. ( z ) (z) ( z ) is OUTSIDE rejection region, we FAIL TO REJECT H 0 H_0 H 0 . There is NOT ENOUGH evidence to conclude that μ = 1200.

Because test stat. ( z ) (z) ( z ) is OUTSIDE rejection region, we FAIL TO REJECT H 0 H_0 H 0 . There is NOT ENOUGH evidence to conclude that μ > 1200.

9. Hypothesis Testing for One Sample

Critical Values and Rejection Regions

9. Hypothesis Testing for One Sample

Critical Values and Rejection Regions: Videos & Practice Problems

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concept

Critical Values and Rejection Regions

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Problem

Mark ‘TRUE’ or ‘FALSE’ for each of the following.
$(A)$ The test statistic & the critical value are the same thing.

TRUE

FALSE

Cannot be determined.

Problem

Mark ‘TRUE’ or ‘FALSE’ for each of the following.
$(B)$ The critical value is the boundary of the rejection region.

TRUE

FALSE

Cannot be determined

Problem

Mark ‘TRUE’ or ‘FALSE’ for each of the following.
$(C)$ You should always reject $H_{0}$ if the test statistic is greater than the critical value.

TRUE

FALSE

Cannot be determined

Problem

Use $α$ to find the critical value(s), then determine if the given test statistic is in the rejection region.
$(A)$ $H_{0} : μ = 7.5; H_{a} : μ > 7.5$
$α = 0.01; z = 2.17$
Test is [ LEFT | TWO | RIGHT ] -tailed
Critical Value(s):
Test stat [ IN | NOT IN ] rejection region.

Test is TWO-tailed, Critical value(s): $\pm2.33$ , Test stat IN rejection region.

Test is LEFT-tailed, Critical value(s): $-2.33$ , Test stat NOT IN rejection region.

Test is RIGHT-tailed, Critical value(s): $2.33$ , Test stat NOT IN rejection region.

Test is RIGHT-tailed, Critical value(s): $2.17$ , Test stat IN rejection region.

Problem

Use $α$ to find the critical value(s), then determine if the given test statistic is in the rejection region.
$(B)$ $H_{0} : p = 0.64; H_{a} : p < 0.64$
$α = 0.10; z = - 1.53$
Test is [ LEFT | TWO | RIGHT ] -tailed
Critical Value(s):
Test stat [ IN | NOT IN ] rejection region.

Test is LEFT-tailed, Critical value(s): $-1.28$ , Test stat NOT IN rejection region.

Test is LEFT-tailed, Critical value(s): $-1.28$ , Test stat IN rejection region.

Test is TWO-tailed, Critical value(s): $\pm1.64$ Test stat IN rejection region.

Test is RIGHT-tailed, Critical value(s): $1.64$ , Test stat IN rejection region.

Problem

Use $α$ to find the critical value(s), then determine if the given test statistic is in the rejection region.
$(C)$ $H_{0} : μ = 98.6$ ; $H_{a} : μ \neq 98.6$
$α = 0.05$
$t = 2.41; d f = 17$
Test is [ LEFT | TWO | RIGHT ] -tailed
Critical Value(s):
Test stat [ IN | NOT IN ] rejection region.

Test is RIGHT-tailed, Critical value(s): $2.11$ , Test stat IN rejection region.

Test is RIGHT-tailed, Critical value(s): $2.11$ , Test stat NOT IN rejection region.

Test is RIGHT-tailed, Critical value(s): $0.025$ , Test stat IN rejection region.

Test is TWO-tailed, Critical value(s): $\pm2.11$ , Test stat IN rejection region.

Problem

Use $α$ to find the critical value(s), then determine if the given test statistic is in the rejection region.
$(D)$ $H_{0} : σ^{2} = 0.3; H_{a} : σ^{2} > 0.3$
$α = 0.05$
$χ^{2} = 61.7; d f = 51$
Test is [ LEFT | TWO | RIGHT ] -tailed
Critical Value(s):
Test stat [ IN | NOT IN ] rejection region.

Test is RIGHT-tailed, Critical value(s): $68.7$ , Test stat IN rejection region.

Test is RIGHT-tailed, Critical value(s): $68.7$ , Test stat NOT IN rejection region.

Test is RIGHT-tailed, Critical value(s): $61.7$ , Test stat NOT IN rejection region.

Test is LEFT-tailed, Critical value(s): $-61.7$ , Test stat NOT IN rejection region.

concept

Performing Hypothesis Test with Critical Values

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Problem

A coffee shop owner believes the shop’s average daily sales are \$1,200, with a known population standard deviation of \$50. A manager claims it’s higher because of their new initiatives & collects a sample of 25 days and finds an average daily sales of \$1,230.
$(A)$ Use $α = 0.10$ & the critical value method to test if the true mean daily sales are higher than \$1200.
$H_{0} : μ =$ _____ $H_{a} : μ$ $\overline{x} =$ $σ =$ $n =$ $z =$
Because test stat. $(z)$ is [ INSIDE | OUTSIDE ] rejection region, we [ REJECT | FAIL TO REJECT ] $H_{0}$ . There is [ ENOUGH | NOT ENOUGH ] evidence to conclude that…

Because test stat. $(z)$ is INSIDE rejection region, we REJECT $H_0$ . There is ENOUGH evidence to conclude that μ = 1200.

Because test stat. $(z)$ is INSIDE rejection region, we REJECT $H_0$ . There is ENOUGH evidence to conclude that μ > 1200.

Because test stat. $(z)$ is OUTSIDE rejection region, we FAIL TO REJECT $H_0$ . There is NOT ENOUGH evidence to conclude that μ = 1200.

Because test stat. $(z)$ is OUTSIDE rejection region, we FAIL TO REJECT $H_0$ . There is NOT ENOUGH evidence to conclude that μ > 1200.

Problem

A coffee shop owner believes the shop’s average daily sales are \$1,200, with a known population standard deviation of \$50. A manager claims it’s higher because of their new initiatives & collects a sample of 25 days and finds an average daily sales of \$1,230.
$(B)$ The owner’s policy is to reward store managers with a yearly bonus for increased sales. Should the owner give this manager with the bonus?

Yes, the manager was able to increase sales.

No, there was no increase in sales.

More info is required.

example

Performing Hypothesis Test with Critical Values Example 1

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Critical Values and Rejection Regions: Videos & Practice Problems

Critical Values and Rejection Regions

Mark ‘TRUE’ or ‘FALSE’ for each of the following.(A)\left(A\right) The test statistic & the critical value are the same thing.

Mark ‘TRUE’ or ‘FALSE’ for each of the following.(B)\left(B\right) The critical value is the boundary of the rejection region.

Mark ‘TRUE’ or ‘FALSE’ for each of the following.(C)\left(C\right) You should always reject H0H_0 if the test statistic is greater than the critical value.

Performing Hypothesis Test with Critical Values

Performing Hypothesis Test with Critical Values Example 1

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Your Statistics for Business tutors

Mark ‘TRUE’ or ‘FALSE’ for each of the following.
$(A)$ The test statistic & the critical value are the same thing.

Mark ‘TRUE’ or ‘FALSE’ for each of the following.
$(B)$ The critical value is the boundary of the rejection region.

Mark ‘TRUE’ or ‘FALSE’ for each of the following.
$(C)$ You should always reject $H_{0}$ if the test statistic is greater than the critical value.