A study on the amount of dye needed to get the best color for a certain type of fabric was conducted. The following data give the photometer readings for the color density of the fabric when the amount of dye used was 1% of the weight of the fabric.
13.2 11.5 12.9 13.0 11.7 10.4 12.1 12.1 11.5 10.3 11.7 12.3 11.2
Do the data demonstrate that the true mean photometer reading for the color density is significantly different from 12 at 3 percent level of significance?
To determine if the data demonstrates that the true mean photometer reading for the color density is significantly different from 12 at the 3 percent level of significance, we can use a hypothesis test.
Let's set up the null and alternative hypotheses:
Null hypothesis (H0): The true mean photometer reading for the color density is 12.
Alternative hypothesis (Ha): The true mean photometer reading for the color density is significantly different from 12.
Next, we need to determine the test statistic to use. Since we don't have the population standard deviation, we can use the t-test. Since the sample size is small (13 readings), we will assume the data is approximately normally distributed.
To calculate the t-test statistic, we need to calculate the sample mean, the sample standard deviation, and the standard error of the mean.
Sample mean (x̄) = sum of all the readings / number of readings
Sample standard deviation (s) = square root of [(sum of (each reading - sample mean)^2) / (number of readings - 1)]
Standard error of the mean (SE) = s / √(number of readings)
Once we have these values, we can calculate the t-test statistic:
t = (x̄ - hypothesized mean) / (SE)
For this case, the hypothesized mean is 12.
Finally, we can compare the t-test statistic to the critical value of the t-distribution at the 3 percent level of significance.
If the absolute value of the t-test statistic is greater than the critical value, we reject the null hypothesis and conclude that the true mean photometer reading for the color density is significantly different from 12. Otherwise, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference.
Note: To find the critical value of the t-distribution, you can use a t-table or a statistical software program.
By following these steps, you can perform the hypothesis test and determine whether the data demonstrate a significant difference.