Answer:
Step 1 of 4
Point estimate for the population mean of the paired differences = -8.2
Step 2 of 4
Sample standard deviation of the paired differences = 16.116244
Step 3 of 4
Margin of Error = ±9.326419
Step 4 of 4
90% Confidence interval = (-17.5, 1.1)
Step-by-step explanation:
The ratings from last year and this year are given in table as
Rating (last year) | x1 | 87 67 68 75 59 60 50 41 75 72
Rating (this year) | x2| 85 52 51 53 50 52 80 44 48 57
Difference | x2 - x1 | -2 -15 -17 -22 -9 -8 30 3 -27 -15
Step 1 of 4
Mean = (Σx)/N = (-82/10) = -8.2 to 1 d.p.
Step 2 of 4
Standard deviation for the sample
= √{[Σ(x - xbar)²]/(N-1)} = 16.116244392951 = 16.116244 to 6 d.p.
Step 3 of 4
Confidence Interval for the population mean is basically an interval of range of values where the true population mean can be found with a certain level of confidence.
Mathematically,
Confidence Interval = (Sample mean) ± (Margin of error)
Sample Mean = -8.2
Margin of Error is the width of the confidence interval about the mean.
It is given mathematically as,
Margin of Error = (Critical value) × (standard Error of the mean)
Critical value will be obtained using the t-distribution. This is because there is no information provided for the population standard deviation.
To find the critical value from the t-tables, we first find the degree of freedom and the significance level.
Degree of freedom = df = n - 1 = 10 - 1 = 9.
Significance level for 90% confidence interval
= (100% - 90%)/2 = 5% = 0.05
t (0.05, 9) = 1.83 (from the t-tables)
Standard error of the mean = σₓ = (σ/√n)
σ = standard deviation of the sample = 16.116244
n = sample size = 10
σₓ = (16.116244/√10) = 5.0964038367
Margin of Error = (Critical value) × (standard Error of the mean) = 1.83 × 5.0964038367 = 9.3264190212 = 9.326419 to 6 d.p.
Step 4 of 4
90% Confidence Interval = (Sample mean) ± (Margin of Error)
CI = -8.2 ± (9.326419)
90% CI = (-17.5264190212, 1.1264190212)
90% Confidence interval = (-17.5, 1.1)
Hope this Helps!!!
