Question

Nevaeh rolled a die 50 times, and 9 of the 50 rolls came up as a six. She wanted to see how likely a result of 9 sixes in 50 rolls would be with a fair die, so Nevaeh used a computer simulation to see the proportion of sixes in 50 rolls, repeated 200 times. Create a 95% confidence interval based on the data from the simulation, to the nearest hundredth, and state whether the observed proportion is within the margin of error of the simulation results.

Answer 1

To create a 95% confidence interval based on the simulation data, we calculate the proportion of sixes in 50 rolls and use the formula for the margin of error. The 95% confidence interval is approximately (0.1132, 0.2468). The observed proportion of sixes (0.18) falls within the confidence interval.

Step-by-step explanation:

To create a 95% confidence interval based on the data from the simulation, we first need to calculate the proportion of sixes in 50 rolls. In this case, we have 9 sixes out of 50 rolls, so the proportion of sixes is 9/50 = 0.18.

Next, we can use the formula for confidence intervals to calculate the margin of error. The margin of error is given by the formula:

Margin of error = z * sqrt((p * (1-p)) / n)

Where z is the z-score corresponding to the desired confidence level (for a 95% confidence level, z is approximately 1.96), p is the proportion of sixes, and n is the number of rolls.

Plugging in the values, we have:

Margin of error = 1.96 * sqrt((0.18 * (1-0.18)) / 50) ≈ 0.0668

Finally, we can construct the confidence interval using the formula:

Confidence interval = 0.18 ± margin of error.

Therefore, the 95% confidence interval is approximately (0.1132, 0.2468).

As for whether the observed proportion is within the margin of error of the simulation results, we can see that the observed proportion of sixes (0.18) does fall within the confidence interval (0.1132, 0.2468). Therefore, we can conclude that the observed proportion is within the margin of error of the simulation results.