A coin was flipped 60 times and came up heads 38 times. At the .10 level of significance, is the coin biased toward heads? (a-1) H0: π ≤ .50

Question

A coin was flipped 60 times and came up heads 38 times. At the .10 level of significance, is the coin biased toward heads? (a-1) H0: π ≤ .50 versus H1: π > .50. Choose the appropriate decision rule at the .10 level of significance. a. Reject H0 if z >1.282 b. Reject H0 if z < 1.282 a b (a-2) Calculate the test statistic. (Carry out all intermediate calculations to at least 4 decimal places. Round your answer to 2 decimal places.) Test statistic (a-3) The null hypothesis should be rejected. True False (a-4) The true proportion is greater than 0.5. True No evidence to support (b-1) Find the p-value. (Round your answer to 4 decimal places.) p-value (b-2) Is the coin biased toward heads? Yes No

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Kennedy 2 weeks 2021-11-19T15:19:45+00:00 1 Answer 0 views 0

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    2021-11-19T15:20:52+00:00

    Answer:

    a1) The null and alternative hypothesis are:

    H_0: \pi=0.5\\\\H_a:\pi>0.5

    Reject H0 if the test statistic is larger than 1.282 (z>1.282)

    a2) Test statistic z=1.94

    a3) True.

    a4) True.

    b1) P-value=0.0264

    b2) Yes, it is biased towards heads.

    Step-by-step explanation:

    This is a hypothesis test for a proportion.

    The claim is that the coin is biased towards heads.

    Then, the null and alternative hypothesis are:

    H_0: \pi=0.5\\\\H_a:\pi>0.5

    The significance level is 0.10.

    The sample has a size n=60.

    The sample proportion is p=0.6333.

    p=X/n=38/60=0.6333

    The standard error of the proportion is:

    \sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.5*0.5}{60}}\\\\\\ \sigma_p=\sqrt{0.00417}=0.0645

    Then, we can calculate the z-statistic as:

    z=\dfrac{p-\pi+0.5/n}{\sigma_p}=\dfrac{0.6333-0.5-0.5/60}{0.0645}=\dfrac{0.125}{0.0645}=1.936

    This test is a right-tailed test, so the P-value for this test is calculated as:

    P-value=P(z>1.936)=0.0264

    As the P-value (0.0264) is smaller than the significance level (0.10), the effect is  significant.

    The null hypothesis is rejected.

    There is  enough evidence to support the claim that the coin is biased towards heads.

    The critical value for a significance level of 0.1 is z=1.282. As this is a right-tailed test, the decision rule is: “Reject H0 if the test statistic is larger than 1.282 (z>1.282)”, as the rejection region is for every z over 1.282.

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