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Acme Confectionery has launched a new bar called Chocolate Crunch - Leaving Cert Mathematics - Question 8 - 2018
Question 8
Acme Confectionery has launched a new bar called Chocolate Crunch. The weights of these new bars are normally distributed with a mean of 64.6 g and a standard deviat... show full transcript
Worked Solution & Example Answer:Acme Confectionery has launched a new bar called Chocolate Crunch - Leaving Cert Mathematics - Question 8 - 2018
Step 1
Find the probability that the mean weight of the sample is between 4.6 g and 4.7 g.
Answer
To find this probability, we will use the Z-score formula:
For the given values:
- Mean ((\mu)) = 64.6 g
- Standard deviation ((\sigma)) = 0.12 g
- Sample size (n) = 10
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Calculating the Z-score for 4.6 g:
- (Z_1 = \frac{4.6 - 64.6}{0.12 / \sqrt{10}} = \frac{-60}{0.0379} = -1581.38)
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Calculating the Z-score for 4.7 g:
- (Z_2 = \frac{4.7 - 64.6}{0.12 / \sqrt{10}} = \frac{-59.9}{0.0379} = -1580.66)
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Use standard normal distribution tables or calculators to find probabilities. The probabilities are insignificantly small, leading to:
- The probability that the mean is between 4.6 g and 4.7 g approaches 0.
Step 2
Create the 95% confidence interval for the population proportion who liked the new bar.
Answer
The sample proportion (\hat{p}) is given by:
Next, calculate the standard error (SE):
Now, for a 95% confidence interval, the Z-score is approximately 1.96. Thus, the margin of error (ME) is:
Now we find the confidence interval:
So,
The 95% confidence interval for the population proportion who liked the new bar is (0.76, 0.86).
Step 3
Put one tick into the table for each statement.
Answer
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When forming confidence intervals (for fixed n and (\hat{p})), an increased confidence level implies a wider interval. -> Always True.
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As the value of (\hat{p}) increases (for fixed n), the estimated standard error of the population proportion increases. -> Sometimes True.
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As the value of (\hat{p}(1 - \hat{p})) increases (for fixed n), the estimated standard error of the population proportion increases. -> Sometimes True.
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As n, the number of people sampled, increases (for fixed (\hat{p})), the estimated standard error of the population proportion increases. -> Never True.
Step 4
Using calculus or otherwise, find the maximum value of \(\hat{p}(1 - \hat{p})\).
Answer
We can express the function as:
To find the maximum value, we differentiate:
Solving for (\hat{p}):
Now, substituting back into the original equation:
Thus, the maximum value of (\hat{p}(1 - \hat{p})) is 0.25.
Step 5
Find the largest possible value of the radius of the 95% confidence interval.
Answer
Using the result from the previous part, we substitute into:
For a sample size of 800:
Hence, the largest possible radius of the 95% confidence interval for a population proportion, given a random sample of size 800, is approximately 0.0347.