Answer :
Final answer:
The t-score is calculated using the sample mean, the population mean under the null hypothesis, the sample's standard deviation, and the sample size. For the provided data, these values lead to a computed t-score of approximately -0.4594.
Explanation:
To calculate the t-score for a set of data when testing a hypothesis, you can use the following formula:
t = (X - \/u0) / (s/√n)
Where X is the sample mean, \/u0 is the population mean according to the null hypothesis, s is the sample standard deviation, and n is the sample size. To find the t-score, you need to know the sample mean (X), the standard deviation (s), and the sample size (n), all of which are usually provided in the problem or can be calculated from the data. In this case:
- X (sample mean) = 98.59
- µ0 (population mean under H0) = 98.6
- s (standard deviation) = 0.0973
- n (sample size) = 20 (number of data points provided)
Plugging these values into the t-score formula gives us:
t = (98.59 - 98.6) / (0.0973/√20)
Perform the calculations:
t = -0.01 / (0.0973/4.4721)
t = -0.01 / 0.02176
t ≈ -0.4594
Thus, the calculated t-score is approximately -0.4594.
Answer:
The t-score is -1.8432
Step-by-step explanation:
We are given the following in the question:
98, 99.6, 97.8, 97.6, 98.7, 98.4, 98.9, 97.1, 99.2, 97.4, 99.1, 96.9, 98.8, 99.9, 96.8, 97, 98.7, 97.6, 98.7, 98.2
Formula:
[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}[/tex]
where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
[tex]Mean =\displaystyle\frac{1964.4}{20} = 98.22[/tex]
Sum of squares of differences = 16.152
[tex]s = \sqrt{\dfrac{16.152}{49}} = 0.922[/tex]
Population mean, μ = 98.6
Sample mean, [tex]\bar{x}[/tex] = 98.22
Sample size, n = 20
Sample standard deviation, s = 0.922
First, we design the null and the alternate hypothesis
[tex]H_{0}: \mu = 98.6\\H_A: \mu < 98.6[/tex]
Formula:
[tex]t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}} }[/tex]
Putting all the values, we have
[tex]t_{stat} = \displaystyle\frac{98.22 - 98.6}{\frac{0.922}{\sqrt{20}} } = -1.8432[/tex]
The t-score is -1.8432