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14 September, 04:06

In a random sample of cars driven at low altitudes, of them exceeded a standard of grams of particulate pollution per gallon of fuel consumed. In an independent random sample of cars driven at high altitudes, of them exceeded the standard. Can you conclude that the proportion of high-altitude vehicles exceeding the standard is less than the proportion of low-altitude vehicles exceeding the standard

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  1. 14 September, 06:44
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    Complete question is;

    In a random sample of 370 cars driven at low altitudes, 43 of them exceeded a standard of 10 grams of particulate pollution per gallon of fuel consumed. In an independent random sample of 80 cars driven at high altitudes, 23 of them exceeded the standard. Can you conclude that the proportion of high-altitude vehicles exceeding the standard is greater than the proportion of low-altitude vehicles exceeding the standard at an level of significance? Group of answer choices

    Answer:

    Yes we can conclude that there is enough evidence to support the claim that the proportion of high-altitude vehicles exceeding the standard is greater than the proportion of low-altitude vehicles exceeding the standard (P-value = 0.00005).

    Step-by-step explanation:

    This is a hypothesis test for the difference between the proportions.

    The claim is that the proportion of high-altitude vehicles exceeding the standard is greater than the proportion of low-altitude vehicles exceeding the standard.

    Then, the null and alternative hypothesis are:

    H0; π1 - π2 = 0

    H1; π1 - π2 < 0

    The significance level would be established in 0.01.

    The random sample 1 (low altitudes), of size n1 = 370 has a proportion of;

    p1 = x1/n1

    p1 = 43/370

    p1 = 0.116

    The random sample 2 (high altitudes), of size n2 = 80 has a proportion of;

    p2 = x2/n2

    p2 = 23/80

    p2 = 0.288

    The difference between proportions is pd = (p1-p2);

    pd = p1 - p2 = 0.116 - 0.288

    pd = - 0.171

    The pooled proportion, we need to calculate the standard error, is:

    p = (x1 + x2) / (n1 + n2)

    p = (43 + 23) / (370 + 80)

    p = 66/450

    p = 0.147

    The estimated standard error of the difference between means is computed using the formula:

    S_ (p1-p2) = √[ ((p (1 - p) / n1) + ((p (1 - p) / n2) ]

    1 - p = 1 - 0.147 = 0.853

    Thus;

    S_ (p1-p2) = √[ ((0.147 * 0.853) / 370) + ((0.147 * 0.853) / 80) ]

    S_ (p1-p2) = 0.044

    Now, we can use the formula for z-statistics as;

    z = (pd - (π1 - π2)) / S_ (p1-p2)

    z = (-0.171 - 0) / 0.044

    z = - 3.89

    Using z-distribution table, we have the p-value = 0.00005

    Since the P-value of (0.00005) is smaller than the significance level (0.01), then the effect is significant.

    We conclude that The null hypothesis is rejected.

    Thus, there is enough evidence to support the claim that the proportion of high-altitude vehicles exceeding the standard is greater than the proportion of low-altitude vehicles exceeding the standard.
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