LeBron’s average bowling score for the season is 180 with a standard deviation of 28. Use technology to determine which of the following represents the probability that LeBron records a score higher than 200.
Responses
76.2% - incorrect
1.1% - incorrect
17% - incorrect
23.8% - correct
explain how you get the correct answer
To find the probability that LeBron records a score higher than 200, we first need to calculate the z-score for a score of 200 using the formula:
z = (X - μ) / σ
where:
X = 200 (score we are interested in)
μ = 180 (mean score)
σ = 28 (standard deviation)
z = (200 - 180) / 28
z = 20 / 28
z = 0.714
Next, we look up the z-score in a standard normal distribution table or use a calculator to find that the probability of scoring higher than 200 corresponds to approximately 23.8% of the distribution.
Therefore, the correct answer is 23.8%.
Given μ=5 and σ=0.5, find the probability that a random variable, x, is between 3.6 and 6.1.
Responses
−98.4% - incorrect
98.6% - incorrect
1.4% - incorrect
98.4% - correct
explain how you get the correct answer
To find the probability that a random variable x is between 3.6 and 6.1, we first need to calculate the z-scores for these values using the formula:
z = (X - μ) / σ
For X = 3.6:
z1 = (3.6 - 5) / 0.5
z1 = -1.4
For X = 6.1:
z2 = (6.1 - 5) / 0.5
z2 = 2.2
Next, we need to find the area under the standard normal distribution curve between these two z-scores (z1 = -1.4 and z2 = 2.2). This can be done using standard normal distribution tables or a calculator.
Using the tables or a calculator, we find that the area between z = -1.4 and z = 2.2 corresponds to approximately 0.984. This means that the probability that x is between 3.6 and 6.1 is 98.4%.
Therefore, the correct answer is 98.4%.
Use the image to answer the following question.
A normal curve is marked and labeled to show the values of a normal distribution. Three percentages are listed above the curve. The Mean and Standard Deviations are below the horizontal axis. Within the curve, 8 regions and percentages are defined. The graph is titled The Empirical Rule.
Given a dataset of 60 values with the normal model N(71, 5) , approximately how many values should fall between 61 and 71 for the normal model to apply?
Responses
48 - incorrect
34 - incorrect
10 - incorrect
29 - correct
explain how you get the correct answer
To answer this question, we can use the Empirical Rule (also known as the 68-95-99.7 Rule) which states that in a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean
- Approximately 95% of the data falls within two standard deviations of the mean
- Approximately 99.7% of the data falls within three standard deviations of the mean
Given the normal model N(71, 5), the values between 61 and 71 are within one standard deviation below the mean. This means that approximately 68% of the data should fall within this range.
Since we are dealing with a dataset of 60 values, 68% of the data falling within one standard deviation of the mean translates to approximately 0.68 * 60 = 40.8 values.
Therefore, approximately 29 values should fall between 61 and 71 for the normal model to apply.
The correct answer is 29.
Lydia collected samples to see how many cars passed by her house in a period of 5 minutes. This is the data that she collected:
17, 8, 17, 21, 16, 15, 13, 10, 18, 17, 23, 17, 18, 10, 18, 13, 16, 21, 17, 16.
The normal model that Lydia calculated for this model is N(16.05, 3.78).
Then, Lydia used the Empirical Rule to check whether this data fits the Empirical Rule. She checked the number of data that were to the left of the +1σ point, which is 19.83. What is the correct conclusion?
A normal curve is marked and labeled to show the values of a normal distribution. Three percentages are listed above the curve. The Mean and Standard Deviations are below the horizontal axis. Within the curve, 8 regions and percentages are defined. The graph is titled The Empirical Rule.
Responses
The normal model is not a good fit because 45% of the data are less than the mean, and the model predicts 50%. - incorrect
The normal model is a good fit because 85% of the data are less than the value at the plus 1 sigma point, and the model predicts 68%. - incorrect
The normal model is not a good fit because 45% of the data are less than the mean, and the model predicts 50%. - incorrect
The normal model is a good fit because 85% of the data are less than the value at the +1σ point, and the model predicts 84%. - correct
explain how you get the correct answer
To determine whether the data fits the Empirical Rule based on the normal model, we need to consider the percentages of data that fall within certain intervals.
The Empirical Rule states that in a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Given Lydia's data has a normal model of N(16.05, 3.78), she checked the number of data points that were to the left of the +1σ point, which is equivalent to finding the percentage of data less than the mean plus one standard deviation. The mean + one standard deviation is 16.05 + 3.78 = 19.83.
Out of the 20 data points Lydia collected, the number of data points that are less than 19.83 is 17. This means that approximately 17/20 = 85% of the data falls below the +1σ point.
The conclusion that the normal model is a good fit because 85% of the data are less than the value at the +1σ point, and the model predicts 84% is correct. This indicates that Lydia's data follows the Empirical Rule as expected based on the normal model she calculated.
Therefore, the correct answer is: The normal model is a good fit because 85% of the data are less than the value at the +1σ point, and the model predicts 84%.