A survey of 50 Leaving Certificate candidates in 2014, randomly selected in the Dublin region, found that they had a mean mark of 374 in a certain subject. The standard deviation of this sample was 45. (a) Find the 95% confidence interval for the mean mark in the subject, in the Dublin region. Interpret this interval. (b) The mean mark in the subject for all Leaving Certificate candidates, in 2014, was 385 and the standard deviation was 45. John suggests that the mean mark in the Dublin region is not the same as in the whole country. Test this hypothesis using a 5% level of significance. Clearly state your null hypothesis, your alternative hypothesis and your conclusion.

Leaving Cert Mathematics Statistics: A survey of 50 Leaving Certifica

A survey of 50 Leaving Certificate candidates in 2014, randomly selected in the Dublin region, found that they had a mean mark of 374 in a certain subject - Leaving Cert Mathematics - Question 1 - 2015

Question 1

A survey of 50 Leaving Certificate candidates in 2014, randomly selected in the Dublin region, found that they had a mean mark of 374 in a certain subject. The stand... show full transcript

Worked Solution & Example Answer:A survey of 50 Leaving Certificate candidates in 2014, randomly selected in the Dublin region, found that they had a mean mark of 374 in a certain subject - Leaving Cert Mathematics - Question 1 - 2015

Step 1

Find the 95% confidence interval for the mean mark in the subject, in the Dublin region.

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Answer

To find the 95% confidence interval for the mean mark, we use the formula:

CI = \left( \bar{x} - z \frac{\sigma}{\sqrt{n}} , \bar{x} + z \frac{\sigma}{\sqrt{n}} \right)

where:

( \bar{x} = 374 ) (mean mark)
( z \approx 1.96 ) (z-score for 95% confidence)
( \sigma = 45 ) (standard deviation)
( n = 50 ) (sample size)

Calculating the margin of error:

ME = 1.96 \times \frac{45}{\sqrt{50}} \approx 12.473

Now substituting in the confidence interval formula:

CI = \left( 374 - 12.473 , 374 + 12.473 \right) = (361.527, 386.473)

Interpretation: We can say with 95% confidence that the mean mark in the subject for candidates in the Dublin area lies in the interval (361.527, 386.473). This means that if we were to repeat this process numerous times, the true mean would likely be within this interval 95% of the time.

Step 2

The mean mark in the subject for all Leaving Certificate candidates, in 2014, was 385 and the standard deviation was 45. John suggests that the mean mark in the Dublin region is not the same as in the whole country. Test this hypothesis using a 5% level of significance.

99%

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Answer

To test John's hypothesis, we will perform a hypothesis test:

Null Hypothesis (H₀): The mean mark in Dublin is the same as the mean mark in the whole country (μ = 385).

Alternative Hypothesis (H₁): The mean mark in Dublin is different from the mean mark in the whole country (μ ≠ 385).

We use a two-tailed z-test for this hypothesis test:

Calculate the test statistic:

Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}

Substituting our values:

Z = \frac{374 - 385}{\frac{45}{\sqrt{50}}} \approx -1.615

Determine the critical z-value for a 5% significance level (two-tailed):

Critical z-values are approximately -1.96 and 1.96.

Conclusion: Since -1.615 is not less than -1.96 and not greater than 1.96, we do not reject the null hypothesis. We do not have sufficient evidence to conclude that the mean mark in Dublin is different from the mean mark in the whole country at the 5% level of significance.

A survey of 50 Leaving Certificate candidates in 2014, randomly selected in the Dublin region, found that they had a mean mark of 374 in a certain subject - Leaving Cert Mathematics - Question 1 - 2015

Worked Solution & Example Answer:A survey of 50 Leaving Certificate candidates in 2014, randomly selected in the Dublin region, found that they had a mean mark of 374 in a certain subject - Leaving Cert Mathematics - Question 1 - 2015

Find the 95% confidence interval for the mean mark in the subject, in the Dublin region.

The mean mark in the subject for all Leaving Certificate candidates, in 2014, was 385 and the standard deviation was 45. John suggests that the mean mark in the Dublin region is not the same as in the whole country. Test this hypothesis using a 5% level of significance.

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