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Answer:
The 95% confidence interval for the mean weight per apple is between 119.14 and 125.86 grams per apple. This means that we are 95% sure that the true mean weight per apple for all the apples they ship is between 119.14 and 125.86 grams per apple.
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1-0.95}{2} = 0.025[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].
So it is z with a pvalue of [tex]1-0.025 = 0.975[/tex], so [tex]z = 1.96[/tex]
Now, find the margin of error M as such
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
[tex]M = 1.96\frac{12}{\sqrt{49}} = 3.36[/tex]
The lower end of the interval is the sample mean subtracted by M. So it is 122.5 - 3.36 = 119.14 grams per apple
The upper end of the interval is the sample mean added to M. So it is 122.5 + 3.36 = 125.86 grams per apple
The 95% confidence interval for the mean weight per apple is between 119.14 and 125.86 grams per apple. This means that we are 95% sure that the true mean weight per apple for all the apples they ship is between 119.14 and 125.86 grams per apple.
The 95% confidence interval for the mean weight per apple is;
CI = (119.14, 125.86)
The interpretation of the confidence interval above is that;
We are 95% sure that the correct mean weight per apple for all the apples they ship is between 119.14 and 125.86 grams per apple.
We are given;
Standard deviation; σ = 12
Sample size; n = 49
Indicated mean weight; x⁻ = 122.5
Now, formula for confidence interval is;
CI = x⁻ ± z(σ/√n)
At Confidence level of 95%, from tables, z = 1.96.
Thus;
CI = 122.5 ± 1.96(12/√49)
CI = 122.5 ± 3.36
CI = (122.5 - 3.36), (122.5 + 3.36)
CI = (119.14, 125.86)
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