Final answer:
The 95% confidence interval for the difference between the means of day and night shift workers' productivity can be calculated using the formula that accounts for the sample means, standard deviations, sizes, and the t-value from the t-distribution, considering the degrees of freedom from the Welch-Satterthwaite equation.
Step-by-step explanation:
To calculate the 95% confidence interval for the difference between the means of day shift and night shift workers' productivity, we will use the formula for the confidence interval of the difference between two independent means, assuming that we do not know the population variances and they are not assumed to be equal (applying Welch's t-test). The formula for the confidence interval is:
CI = (mean1 - mean2) ± t* ∙ √((s1^2/n1) + (s2^2/n2))
Where:
- mean1 and mean2 are the sample means for day and night shifts, respectively.
- s1 and s2 are the sample standard deviations.
- n1 and n2 are the sample sizes.
- t* is the t-value from the t-distribution for the desired confidence level and degrees of freedom calculated using Welch-Satterthwaite equation.
For the day shift sample (mean1 = 68, s1 = 15, n1 = 42) and night shift sample (mean2 = 63, s2 = 8, n2 = 33), we can substitute these values into the formula. The t-value for a 95% confidence interval can be found using a t-distribution table or software with the appropriate degrees of freedom from the Welch-Satterthwaite equation. The degrees of freedom can be estimated using the formula:
df = ((s1^2/n1 + s2^2/n2)^2) / (((s1^2/n1)^2/(n1-1)) + ((s2^2/n2)^2/(n2-1)))
Once we have the t-value and degrees of freedom, we can calculate the confidence interval's range.