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Pennies made prior to 1982 were made of 95% copper. Because of their copper content, these pennies are worth about $0.023 each. Pennies made after 1982 are only 2.5% copper. Jenna reads online that 13.2% of pennies in circulation are pre-1982 copper pennies. Jenna has a large container of pennies at home. She selects a random sample of 50 pennies from the container and finds that 11 are pre-1982 copper pennies. Does this provide convincing evidence that the proportion of pennies in her containers that are pre-1982 copper pennies is greater than 0.132

Sagot :

Using the z-distribution, it is found that since the test statistic is less than the critical value for the right-tailed test, it is found that this does not provide convincing evidence that the proportion of pennies in her containers that are pre-1982 copper pennies is greater than 0.132.

At the null hypothesis, it is tested if the proportion of pennies in her containers that are pre-1982 copper pennies not greater than 0.132, that is:

[tex]H_0: p \leq 0.132[/tex]

At the alternative hypothesis, it is tested if it is greater, that is:

[tex]H_1: p > 0.132[/tex]

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 = 0.132, n = 50, \overline{p} = \frac{11}{50} = 0.22[/tex]

Hence, the value of the test statistic is given by:

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

[tex]z = \frac{0.22 - 0.132}{\sqrt{\frac{0.5(0.5)}{50}}}[/tex]

[tex]z = 1.24[/tex]

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

Since the test statistic is less than the critical value for the right-tailed test, it is found that this does not provide convincing evidence that the proportion of pennies in her containers that are pre-1982 copper pennies is greater than 0.132.

You can learn more about the use of the z-distribution to test an hypothesis at https://brainly.com/question/16313918

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