Question: Top Rated TV Programs: 2007-2008\t\tTop Rated TV Programs: 2012-2013\nProgram\tRating\tProgram\tRating\nDancing With the Stars - Tuesday (2008)\t$11.8$\tAmerican Idol - Wednesday\t$4.6$\nDesperate Housewives\t$11.6$\tAmerican Idol - Thursday\t$4.3$\nCSI\t$10.6$\tThe Following\t$4.3$\nHouse - Tuesday\t$10.5$\tTwo and a Half Men\t$4.1$\nGrey's Anatomy\t$10.4$\tGrey's Anatomy\t$4.1$\nSunday Night Football\t$9.7$\tNCIS\t$4$\nCSI: Miami\t$9.2$\tFootball Night in America Part 3\t$4$\nHouse - Monday\t$9.2$\tRevolution\t$3.9$\nstandard deviations for 2007-2008 and 2012-2013

1. **State the problem:** Calculate the standard deviations of the TV program ratings for the years 2007-2008 and 2012-2013. 2. **Recall the formula for standard deviation:** $$\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^n (x_i - \mu)^2}$$ where $\mu$ is the mean of the data, $x_i$ are the data points, and $n$ is the number of data points. 3. **Calculate the mean for 2007-2008:** Ratings: $11.8, 11.6, 10.6, 10.5, 10.4, 9.7, 9.2, 9.2$ $$\mu_{07-08} = \frac{11.8 + 11.6 + 10.6 + 10.5 + 10.4 + 9.7 + 9.2 + 9.2}{8} = \frac{82.99999999999999}{8} = 10.375$$ 4. **Calculate each squared deviation for 2007-2008:** $$ (11.8 - 10.375)^2 = 2.052625$$ $$ (11.6 - 10.375)^2 = 1.496025$$ $$ (10.6 - 10.375)^2 = 0.050625$$ $$ (10.5 - 10.375)^2 = 0.015625$$ $$ (10.4 - 10.375)^2 = 0.000625$$ $$ (9.7 - 10.375)^2 = 0.455625$$ $$ (9.2 - 10.375)^2 = 1.380625$$ $$ (9.2 - 10.375)^2 = 1.380625$$ 5. **Sum the squared deviations for 2007-2008:** $$2.052625 + 1.496025 + 0.050625 + 0.015625 + 0.000625 + 0.455625 + 1.380625 + 1.380625 = 6.832375$$ 6. **Calculate the variance and standard deviation for 2007-2008:** $$\sigma^2_{07-08} = \frac{6.832375}{8} = 0.854046875$$ $$\sigma_{07-08} = \sqrt{0.854046875} \approx 0.924$$ 7. **Calculate the mean for 2012-2013:** Ratings: $4.6, 4.3, 4.3, 4.1, 4.1, 4, 4, 3.9$ $$\mu_{12-13} = \frac{4.6 + 4.3 + 4.3 + 4.1 + 4.1 + 4 + 4 + 3.9}{8} = \frac{33.3}{8} = 4.1625$$ 8. **Calculate each squared deviation for 2012-2013:** $$ (4.6 - 4.1625)^2 = 0.19140625$$ $$ (4.3 - 4.1625)^2 = 0.01890625$$ $$ (4.3 - 4.1625)^2 = 0.01890625$$ $$ (4.1 - 4.1625)^2 = 0.00390625$$ $$ (4.1 - 4.1625)^2 = 0.00390625$$ $$ (4 - 4.1625)^2 = 0.02640625$$ $$ (4 - 4.1625)^2 = 0.02640625$$ $$ (3.9 - 4.1625)^2 = 0.06890625$$ 9. **Sum the squared deviations for 2012-2013:** $$0.19140625 + 0.01890625 + 0.01890625 + 0.00390625 + 0.00390625 + 0.02640625 + 0.02640625 + 0.06890625 = 0.35875$$ 10. **Calculate the variance and standard deviation for 2012-2013:** $$\sigma^2_{12-13} = \frac{0.35875}{8} = 0.04484375$$ $$\sigma_{12-13} = \sqrt{0.04484375} \approx 0.212$$ **Final answers:** - Standard deviation for 2007-2008 ratings is approximately $0.924$. - Standard deviation for 2012-2013 ratings is approximately $0.212$.