9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Proportions

9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Proportions

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Multiple Choice

Perform a 2-tailed hypothesis test for the true proportion of successes using the given values:
$alpha equals 0.10$ , $n equals 100$ , $x equals 42$ , & claim is $p equals 0.25$

Because $P$ -value = 0.00004 < $\alpha=$ 0.01, we FAIL TO REJECT $H_0$ . There is NOT ENOUGH evidence to suggest $H_{a}$ : $p$ ≠ 0.75

Because $P$ -value = 0.00004 < $\alpha=$ 0.01, we REJECT $H_0$ . There is ENOUGH evidence to suggest $H_{a}$ : $p$ ≠ 0.75

Because $P$ -value = 0.00008 < $\alpha=$ 0.01, we REJECT $H_0$ . There is ENOUGH evidence to suggest $H_{a}$ : $p$ ≠ 0.75

Because $P$ -value = 0.00008 < $\alpha=$ 0.01, we FAIL TO REJECT $H_0$ . There is NOT ENOUGH evidence to suggest $H_{a}$ : $p$ ≠ 0.75

Verified step by step guidance

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is H₀: p = 0.25, and the alternative hypothesis is Hₐ: p ≠ 0.25, since this is a two-tailed test.

Step 2: Calculate the sample proportion (p̂). The formula for p̂ is p̂ = x / n, where x is the number of successes (42) and n is the sample size (100).

Step 3: Verify the conditions for performing a hypothesis test for proportions. Ensure that both np and nq are greater than or equal to 5, where q = 1 - p. For np, calculate np = n * p, and for nq, calculate nq = n * q.

Step 4: Compute the test statistic (z-value). The formula for the z-value is z = (p̂ - p) / √(p * q / n), where p̂ is the sample proportion, p is the claimed proportion, q = 1 - p, and n is the sample size.

Step 5: Compare the calculated z-value to the critical z-value for α = 0.10 (two-tailed test) or use the p-value approach. If the p-value is less than α, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

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Related Practice

Textbook Question

True or False: Sample evidence can prove a null hypothesis is true.

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Open Question

A city government claims that no more than $25 percent sign$ of households have solar panels. A researcher suspects the rate is actually higher and surveys $200$ households, finding that $62$ have solar panels. Test if there is evidence that more than $25 percent sign$ of households have solar panels using $alpha equals 0.01$ .

$H sub 0 colon$ _________ $H_{a} :$ _________

$x =$ ________ $n =$ ________ $P$ -value: ______

Because $P$ -value [ < | > ] $α$ , we [ REJECT | FAIL TO REJECT ] $H sub 0$ , there is [ ENOUGH | NOT ENOUGH ] evidence to suggest…

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Open Question

A city claims that $30 percent sign$ of households recycle regularly. A researcher surveys $200$ households to see if the true proportion is different and finds that $78$ recycle regularly. Use $alpha equals 0.05$ to test the claim.

$H sub 0 colon$ _________ $H_{a} :$ _________

$x =$ ________ $n =$ ________ $P$ -value:______

Because $P$ -value [ < | > ] $α$ , we [ REJECT | FAIL TO REJECT ] $H sub 0$ , there is [ ENOUGH | NOT ENOUGH ] evidence to suggest…

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Multiple Choice

Perform a 2-tailed hypothesis test for the true proportion of successes using the given values:

$alpha equals 0.01$ , $n equals 40$ , $x equals 28$ , & claim is $p equals 0.75$

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Multiple Choice

A local park claims that less than 15% of visitors litter. A random sample of 120 visitors finds that 25 litter. At the 0.05 significance level, test if the proportion of visitors who litter is greater than 15%.

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Textbook Question

Feeling Your Age A research organization conducts a survey by randomly selecting adults and asking each, “How do you feel relative to your age?” The results are shown in the figure. (Adapted from Pew Research Center)

[IMAGE]

a. Use a sign test to test the null hypothesis that the proportion of adults who feel older is equal to the proportion of adults who feel younger. Assign a + sign to each adult who responded “older,” assign a - sign to each adult who responded “younger,” and assign a 0 to each adult who responded “my age.” Use α = 0.05

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Textbook Question

Contacting Parents A research organization conducts a survey by randomly selecting adults and asking each, “How frequently do you contact your parents by phone?” The results are shown in the figure. (Adapted from Pew Research Center)

a. Use a sign test to test the null hypothesis that the proportion of adults who contact their parents by phone weekly is equal to the proportion of adults who contact their parents by phone daily. Assign a + sign to each adult who responded “weekly,” assign a - sign to each adult who responded “daily,” and assign a 0 to each adult who responded “other.” Use α = 0.05

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Textbook Question

Explain how to test a population proportion p.

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Performing Hypothesis Tests: Proportions practice set

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Struggling with Statistics?

Perform a 2-tailed hypothesis test for the true proportion of successes using the given values:α=0.10α=0.10, n=100n=100, x=42x=42, & claim is p=0.25p=0.25

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Performing Hypothesis Tests: Proportions practice set

Perform a 2-tailed hypothesis test for the true proportion of successes using the given values:
$alpha equals 0.10$ , $n equals 100$ , $x equals 42$ , & claim is $p equals 0.25$