The diameter of a ball bearing is to have an averageof 2.0 Cm. A random sampling of 10 bearings is selected every hour to check the diameters. The sample selected at 9 a.m gave the following measurements: 2.02, 2.04, 2.00, 1.98, 2.04, 2.04, 2.00, 2.02, 1.96. Calculate the t-value and draw your conclusions,if any.
To calculate the t-value, we need to follow these steps:
Step 1: Compute the sample mean.
The sample mean (x̄) is obtained by summing all the measurements and dividing by the number of measurements:
x̄ = (2.02 + 2.04 + 2.00 + 1.98 + 2.04 + 2.04 + 2.00 + 2.02 + 1.96) / 9 = 19.10 / 9 = 2.122
Step 2: Compute the sample standard deviation.
The sample standard deviation (s) measures the spread of the data points around the sample mean. To calculate it, we need to subtract the sample mean from each measurement, square the differences, sum them up, divide by (n-1), and then take the square root:
s = √[(∑(x - x̄)²) / (n - 1)]
= √[((2.02 - 2.122)² + (2.04 - 2.122)² + ... + (1.96 - 2.122)²) / (9 - 1)]
= √[(0.0084 + 0.0064 + ... + 0.0164) / 8]
= √[0.0988 / 8]
= √0.01235
= 0.1110
Step 3: Calculate the standard error.
The standard error (SE) measures the uncertainty of the sample mean. It is calculated by dividing the sample standard deviation by the square root of the sample size:
SE = s / √n
SE = 0.1110 / √9
SE = 0.0370
Step 4: Calculate the t-value.
The t-value (t) represents the difference between the sample mean and the population mean (2.0 cm) divided by the standard error:
t = (x̄ - μ) / SE
= (2.122 - 2.0) / 0.0370
= 0.122 / 0.0370
= 3.297
The t-value for this sample is 3.297.
Step 5: Interpretation.
To draw conclusions from the t-value, we need to compare it to the critical t-value at the desired significance level (alpha). If the calculated t-value is greater than the critical t-value, we reject the null hypothesis. If it is smaller, we fail to reject the null hypothesis.
Since the question does not mention an alpha level or provide the critical t-value, we cannot determine whether the calculated t-value leads to rejecting or failing to reject the null hypothesis.
Note: The null hypothesis, in this case, would be that the average diameter of the ball bearings is 2.0 cm. The alternative hypothesis would be that the average diameter is different from 2.0 cm.