Are the new quarters fair? Flip 5 different coins 50x times. Data : coin#1 CA 20

Question

Are the new quarters fair? Flip 5 different coins 50x times.

Data :
coin#1 CA 2016 29 Heads 21 Tails
coin #2 VA 2015 27 Heads 23 Tails
coin #3 AL 2009 31 Heads 19 Tails
coin#4 NV 2000. 23 Heads 27 Tails
coin #5 OH 1999 28 heads 22 Tails

For each coin, perform a hypothesis test.
Which hypothesis test is appropriate?
Are the conditions met?
State your null and alternative hypotheses. Calculate the test statistics.
Using a significance level of a=0.05, decide whether to reject or not reject the null hypothesis.
What is your conclusion?

Explanation / Answer

Solution:-

1) Coin 1

n1p1> 10

n1(1 - p1) > 10

Hence all conditions are met for two proportion z-test.

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: PHead = PTail

Alternative hypothesis: PHead PTail

Note that these hypotheses constitute a two-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p1 * n1 + p2 * n2) / (n1 + n2)

p = 0.50

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }

SE = 0.071

z = (p1 - p2) / SE

z = 2.25

where p1 is the sample proportion in sample 1, where p2 is the sample proportion in sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.

Since we have a two-tailed test, the P-value is the probability that the z-score is less than -2.25 or greater than 2.25.

Thus, the P-value = 0.0244

Interpret results. Since the P-value (0.0244) is less than the significance level (0.05), we cannot accept the null hypothesis.

2) Coin 2

n2p2> 10

n2(1 - p2) > 10

Hence all conditions are met for two proportion z-test.

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: PHead = PTail

Alternative hypothesis: PHead PTail

Note that these hypotheses constitute a two-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p1 * n1 + p2 * n2) / (n1 + n2)

p = 0.50

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }

SE = 0.071

z = (p1 - p2) / SE

z = 1.13

where p1 is the sample proportion in sample 1, where p2 is the sample proportion in sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.

Since we have a two-tailed test, the P-value is the probability that the z-score is less than - 1.13 or greater than 1.13

Thus, the P-value = 0.2584

Interpret results. Since the P-value (0.2584) is greater than the significance level (0.05), we have to accept the null hypothesis.

3) Coin 3

n3p3> 10

n3(1 - p3) > 10

Hence all conditions are met for two proportion z-test.

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: PHead = PTail

Alternative hypothesis: PHead PTail

Note that these hypotheses constitute a two-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p1 * n1 + p2 * n2) / (n1 + n2)

p = 0.50

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }

SE = 0.071

z = (p1 - p2) / SE

z = 3.38

where p1 is the sample proportion in sample 1, where p2 is the sample proportion in sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.

Since we have a two-tailed test, the P-value is the probability that the z-score is less than - 3.38 or greater than 3.38.

Thus, the P-value = 0.0008

Interpret results. Since the P-value (0.0008) is less than the significance level (0.05), we have to reject the null hypothesis.

4)

Coin 4

n4p4> 10

n4(1 - p4) > 10

Hence all conditions are met for two proportion z-test.

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: PHead = PTail

Alternative hypothesis: PHead PTail

Note that these hypotheses constitute a two-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p1 * n1 + p2 * n2) / (n1 + n2)

p = 0.50

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }

SE = 0.071

z = (p1 - p2) / SE

z = - 1.13

where p1 is the sample proportion in sample 1, where p2 is the sample proportion in sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.

Since we have a two-tailed test, the P-value is the probability that the z-score is less than - 1.13 or greater than 1.13

Thus, the P-value = 0.2584

Interpret results. Since the P-value (0.2584) is greater than the significance level (0.05), we have to accept the null hypothesis.

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