Among a simple random sample of 331 American adults who do not have a four-year college degree and are not currently enrolled in school, 48%

Question

Among a simple random sample of 331 American adults who do not have a four-year college degree and are not currently enrolled in school, 48% said they decided not to go to college because they could not afford school.49 A newspaper article states that only a minority of the Americans who decide not to go to college do so because they cannot afford it and uses the point estimate from this survey as evidence. Conduct a hypothesis test to determine if these data provide strong evidence supporting this statement using the 7 steps and a significance level of 0.05.

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Sophia 2 weeks 2021-10-14T01:58:12+00:00 1 Answer 0

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    2021-10-14T01:59:59+00:00

    Answer:

    Step-by-step explanation:

    We want to test hypothesis:

    H_0 : p\geq0.5

    H_a : p < 0.5

    This is a lower tailed test

    Sample proportion. \stackrel{\wedge}{p}=0.48, n= 331

    And claimed proportion, P=0.5

    Significance level, α=0.05  (if no value is given, we take level of 0.05)

    Now, calculating statistics

    Standard deviation of \stackrel{\wedge}{p},\sigma_{\stackrel{\wedge}{p}} = \frac{\sqrt{P*(1-P)}}{n}

    = \frac{\sqrt{0.05*(1-0.05)}}{331}

     \approx0.0275

    Test statistic,z_{observed }= \frac{(\stackrel{\wedge}{p}-0.5)}{ \sigma_{\stackrel{\wedge}{p}}}

    =\frac{ ((0.48)-0.5)}{0.0275}

    \approx -0.727736

    \approx  -0.73

    Test statistic: -0.73

    Since this is lower tailed test, p value = P(Z < z_{observed})= P ( Z < -0.73) = 0.2327

    p-value= 0.2327

    note that exact p-value is : 0.2333875382

    Rejection criteria : reject H0 if p-value < \alpha

    Decision: Sincep-value\geq\alpha, we fail to reject the null hypothesis. There is insufficient evidence to conclude that p is less than 0.5

    Alternatively, we can use critical value approach,

    Z_c= -z_\alpha=-z_{0.05}=-z_{0.05}=-1.645(From z table, using interpolation, ½th distance between -1.64 and -1.65)

    critical value = -1.645

    Rejection criteria: RejectH_0 if z_0< -1.645

    Decision : since z_o \geq z_c, we fail to reject the null hypothesis. There is insufficient evidence to conclude that p is less than to 0.5

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