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Chapel Hill Carrboro City Schools claim that 82% of students pass their EOY tests but a survey of 100 randomly selected
local students who had taken the test found only 75 who passed. Does this finding suggest that the passing rate for
teenagers is lower than what CHCCS reported?
Step 1: Hypotheses
Null:
Alternative:
Step 2: Check Conditions
Randomization:
10% condition:
Success/failure condition:
Step 3: Mechanics:
Step 4: Stating the conclusion:

Sagot :

Using the z-distribution, it is found that since the test statistic is less than the critical value for the left-tailed test, this finding does suggest that the passing rate for  teenagers is lower than what CHCCS reported.

Hypothesis

  • At the null hypothesis, it is tested if the passing rate is of at least 82%, that is:

[tex]H_0: p \geq 0.82[/tex]

  • At the alternative hypothesis, it is tested if the passing rate is of less than 82%, that is:

[tex]H_1: p < 0.82[/tex]

Randomiztion and 10% condition

  • The students on the sample are chosen at random, hence the randomization condition is respected.
  • 75 students passed, 25 did not, hence there are at least 10 successes and 10 failures in the sample and the 10% condition is also respected.

Test statistic

The test statistic is given by:

[tex]z = \frac{\overline{p} - p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]

In which:

  • [tex]\overline{p}[/tex] is the sample proportion.
  • p is the proportion tested at the null hypothesis.
  • n is the sample size.

In this problem, the parameters are:

[tex]p = 0.82, n = 100, \overline{p} = \frac{75}{100} = 0.75[/tex]

Hence, the value of the test statistic is:

[tex]z = \frac{\overline{p} - p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]

[tex]z = \frac{0.75 - 0.82}{\sqrt{\frac{0.82(0.18)}{100}}}[/tex]

[tex]z = -1.82[/tex]

The critical value for a left-tailed test, as we are testing if the proportion is less than a value, with a significance level of 0.05 is of [tex]z^{\ast} = -1.645[/tex]

Since the test statistic is less than the critical value for the left-tailed test, this finding does suggest that the passing rate for  teenagers is lower than what CHCCS reported.

To learn more about the z-distribution, you can take a look at https://brainly.com/question/26013190

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