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Answer :
The test statistic, calculated using the sample mean and standard deviation for a one-sample t-test, is approximately -2.41. This is computed using the formula for the t-test statistic, considering the null hypothesis that the population mean is 98.6°F.
To find the value of the test statistic, we must perform a one-sample t-test since the population standard deviation is unknown, and we are comparing the sample mean to a known value. For this test, the null hypothesis (
H0) is that the mean body temperature of healthy adults is 98.6°F while the alternative hypothesis (
H1) is that the mean body temperature is less than 98.6°F.
To calculate the t-test statistic, we use the formula:
t = (x' - μ) / (s / √n)
Where:
- x' is the sample mean,
- μ is the population mean (98.6°F),
- s is the sample standard deviation, and
- n is the sample size.
First, we calculate the sample mean (x') as: x' = (98.2 + 98.7 + 98.7 + 98.4 + 97.2 + 98.1) / 6 = 589.3 / 6 = 98.22°F
Now, we find the sample standard deviation (s):
s = √[Σ(x - x')² / (n-1)]
Where x represents each data point. Calculating this...
s = √[(98.2-98.22)² + (98.7-98.22)² + (98.7-98.22)² + (98.4-98.22)² + (97.2-98.22)² + (98.1-98.22)² / 5]
s ≈ √[0.0004 + 0.2304 + 0.2304 + 0.0324 + 1.0404 + 0.0144 / 5]
s ≈ √[1.548 / 5]
s ≈ √[0.3096]
s ≈ 0.5565°F
Using the sample standard deviation and mean, we find the t-test statistic:
t = (98.22 - 98.6) / (0.5565 / √6) ≈ -2.41
Therefore, the value of the t-test statistic -2.41.
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Final answer:
To find the value of the test statistic, calculate the sample mean and sample standard deviation. Then use these values to calculate the test statistic.
Explanation:
To find the value of the test statistic, we first need to calculate the sample mean. The sample mean can be found by adding up all the temperatures and dividing by the number of temperatures.
Sum of temperatures: 98.2 + 98.7 + 98.7 + 98.4 + 97.2 + 98.1 = 589.3
Number of temperatures: 6
Sample mean: 589.3/6 = 98.2167
Next, we need to calculate the sample standard deviation. The sample standard deviation measures the spread of the data.
Step 1: Calculate the deviations for each temperature by subtracting the sample mean from each temperature.
Deviation for 98.2: 98.2 - 98.2167 = -0.0167
Deviation for 98.7: 98.7 - 98.2167 = 0.4833
Deviation for 98.7: 98.7 - 98.2167 = 0.4833
Deviation for 98.4: 98.4 - 98.2167 = 0.1833
Deviation for 97.2: 97.2 - 98.2167 = -1.0167
Deviation for 98.1: 98.1 - 98.2167 = -0.1167
Step 2: Square each deviation.
Squared deviation for -0.0167: (-0.0167)^2 = 0.00027789
Squared deviation for 0.4833: (0.4833)^2 = 0.23380889
Squared deviation for 0.4833: (0.4833)^2 = 0.23380889
Squared deviation for 0.1833: (0.1833)^2 = 0.03361289
Squared deviation for -1.0167: (-1.0167)^2 = 1.03345689
Squared deviation for -0.1167: (-0.1167)^2 = 0.01361089
Step 3: Calculate the mean of the squared deviations.
Mean of squared deviations: (0.00027789 + 0.23380889 + 0.23380889 + 0.03361289 + 1.03345689 + 0.01361089)/6 = 0.225721084
Step 4: Take the square root of the mean squared deviation to find the standard deviation.
Sample standard deviation: sqrt(0.225721084) = 0.47489
Finally, we can use the sample mean and sample standard deviation to calculate the test statistic.
Test statistic = (sample mean - population mean)/(sample standard deviation/sqrt(sample size))
Test statistic = (98.2167 - 98.6)/(0.47489/sqrt(6)) = -0.3833/(0.47489/2.4495) = -0.3833/0.1942 = -1.9747
The value of the test statistic is -1.9747.
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