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Answer :
To calculate the test statistic for this problem, we use the provided formula for the test statistic [tex]\( z_0 \)[/tex]:
[tex]\[ z_0 = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}} \][/tex]
Let's break down each part of the formula using the given values:
- [tex]\(\bar{x}\)[/tex] is the sample mean, which is 99.8 degrees Fahrenheit.
- [tex]\(\mu_0\)[/tex] is the population mean under the null hypothesis, which is 98.6 degrees Fahrenheit.
- [tex]\(\sigma\)[/tex] is the population standard deviation, known to be 15 degrees Fahrenheit.
- [tex]\(n\)[/tex] is the sample size, which is 40.
Now, substitute these values into the formula:
1. Calculate the difference between the sample mean and the population mean:
[tex]\(\bar{x} - \mu_0 = 99.8 - 98.6 = 1.2\)[/tex]
2. Calculate the standard error of the mean, which involves dividing the population standard deviation by the square root of the sample size:
[tex]\(\frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{40}}\)[/tex]
3. Calculate the test statistic [tex]\( z_0 \)[/tex]:
[tex]\[ z_0 = \frac{1.2}{\frac{15}{\sqrt{40}}} \][/tex]
4. Simplify the calculation to find [tex]\( z_0 \)[/tex]:
Perform the division:
[tex]\(\frac{15}{\sqrt{40}} \approx 2.3717\)[/tex]
Now, divide the difference in means by the standard error:
[tex]\[ z_0 = \frac{1.2}{2.3717} \approx 0.5059644256269419 \][/tex]
5. Finally, round [tex]\( z_0 \)[/tex] to two decimal places to get the final answer:
[tex]\( z_0 \approx 0.51 \)[/tex]
So, the test statistic [tex]\( z_0 \)[/tex] is approximately 0.51.
[tex]\[ z_0 = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}} \][/tex]
Let's break down each part of the formula using the given values:
- [tex]\(\bar{x}\)[/tex] is the sample mean, which is 99.8 degrees Fahrenheit.
- [tex]\(\mu_0\)[/tex] is the population mean under the null hypothesis, which is 98.6 degrees Fahrenheit.
- [tex]\(\sigma\)[/tex] is the population standard deviation, known to be 15 degrees Fahrenheit.
- [tex]\(n\)[/tex] is the sample size, which is 40.
Now, substitute these values into the formula:
1. Calculate the difference between the sample mean and the population mean:
[tex]\(\bar{x} - \mu_0 = 99.8 - 98.6 = 1.2\)[/tex]
2. Calculate the standard error of the mean, which involves dividing the population standard deviation by the square root of the sample size:
[tex]\(\frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{40}}\)[/tex]
3. Calculate the test statistic [tex]\( z_0 \)[/tex]:
[tex]\[ z_0 = \frac{1.2}{\frac{15}{\sqrt{40}}} \][/tex]
4. Simplify the calculation to find [tex]\( z_0 \)[/tex]:
Perform the division:
[tex]\(\frac{15}{\sqrt{40}} \approx 2.3717\)[/tex]
Now, divide the difference in means by the standard error:
[tex]\[ z_0 = \frac{1.2}{2.3717} \approx 0.5059644256269419 \][/tex]
5. Finally, round [tex]\( z_0 \)[/tex] to two decimal places to get the final answer:
[tex]\( z_0 \approx 0.51 \)[/tex]
So, the test statistic [tex]\( z_0 \)[/tex] is approximately 0.51.
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