Question

during a school carnival, students played a ring toss game. the data set below shows the number of rings successfully tossed onto the target by each student:
\\{8, 6, 10, 9, 7, 8, 6, 9, 7, 11, 8, 10, 6, 7, 8\\}. what is the probability that a student tossed at least 8 rings onto the target during the carnival?

use the geogebra spreadsheet to find the mean and standard deviation. use these values to create a normal distribution for the data set. then use the geogebra probability calculator to solve the problem.

type your answer in the space provided. (use two decimal places only if your answer is not an exact whole number.)

the probability that a student successfully tossed at least 8 rings is , with a mean of rings and standard deviation of rings.

Explanation:

Calculate the sample mean

Using the Sample Mean knowledge point, we sum the data values and divide by the sample size \(n = 16\).

$$ \sum x = 8+6+10+9+7+8+6+9+7+11+8+10+6+7+8 = 120 $$

$$ \mu = \frac{120}{16} = 7.5 $$

Calculate the sample standard deviation

Using the Sample Standard Deviation knowledge point, we compute the sample standard deviation \(s\).

$$ \sum x^2 = 8^2+6^2+10^2+9^2+7^2+8^2+6^2+9^2+7^2+11^2+8^2+10^2+6^2+7^2+8^2 = 938 $$

$$ s = \sqrt{\frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n-1}} = \sqrt{\frac{938 - \frac{120^2}{16}}{15}} = \sqrt{\frac{938 - 900}{15}} = \sqrt{\frac{38}{15}} \approx 1.59 $$

Compute the normal probability

Using the Normal Distribution Probability knowledge point, we find \(P(X \ge 8)\) for \(X \sim N(7.5, 1.59^2)\).

$$ z = \frac{8 - 7.5}{1.5921} \approx 0.314 $$

$$ P(Z \ge 0.31) \approx 0.38 $$

Answer:

The probability that a student successfully tossed at least 8 rings is <blank>0.38</blank>, with a mean of <blank>7.5</blank> rings and standard deviation of <blank>1.59</blank> rings.