Answer :
To determine what percent of outcomes in a standard normal distribution occur within 1.03 standard deviations from the mean, we need to follow these steps:
1. Understand the standard normal distribution:
- The standard normal distribution is a bell-shaped curve that is symmetrical around the mean (which is 0). In this distribution, the spread is measured by the standard deviation (σ), and the total area under the curve represents all possible outcomes (100%).
2. Use the Z-score:
- The Z-score represents the number of standard deviations a data point is from the mean. In this case, we are looking at ±1.03 standard deviations from the mean (0).
3. Find the cumulative probability for Z = 1.03:
- The cumulative probability for a Z-score gives us the probability that a value is less than or equal to that Z-score. For a Z-score of 1.03, the cumulative probability (the area under the curve to the left of 1.03) is about 0.8485.
4. Symmetry of the distribution:
- Due to the symmetry of the normal distribution, the cumulative probability for -1.03 is the same as that for 1.03, but it captures the area to the left of -1.03. This area is approximately 0.1515 (since 1 - 0.8485 = 0.1515).
5. Calculate the area within ±1.03:
- To find the probability of being within ±1.03 standard deviations, subtract the cumulative probability for -1.03 from that for 1.03:
[tex]\[ P(-1.03 \leq Z \leq 1.03) = P(Z \leq 1.03) - P(Z \leq -1.03) \approx 0.8485 - 0.1515 = 0.6970 \][/tex]
6. Convert to percentage:
- To express this probability as a percentage, multiply by 100:
[tex]\[ 0.6970 \times 100 = 69.7\% \][/tex]
Therefore, approximately 69.7% of outcomes in a standard normal distribution occur within 1.03 standard deviations from the mean.
1. Understand the standard normal distribution:
- The standard normal distribution is a bell-shaped curve that is symmetrical around the mean (which is 0). In this distribution, the spread is measured by the standard deviation (σ), and the total area under the curve represents all possible outcomes (100%).
2. Use the Z-score:
- The Z-score represents the number of standard deviations a data point is from the mean. In this case, we are looking at ±1.03 standard deviations from the mean (0).
3. Find the cumulative probability for Z = 1.03:
- The cumulative probability for a Z-score gives us the probability that a value is less than or equal to that Z-score. For a Z-score of 1.03, the cumulative probability (the area under the curve to the left of 1.03) is about 0.8485.
4. Symmetry of the distribution:
- Due to the symmetry of the normal distribution, the cumulative probability for -1.03 is the same as that for 1.03, but it captures the area to the left of -1.03. This area is approximately 0.1515 (since 1 - 0.8485 = 0.1515).
5. Calculate the area within ±1.03:
- To find the probability of being within ±1.03 standard deviations, subtract the cumulative probability for -1.03 from that for 1.03:
[tex]\[ P(-1.03 \leq Z \leq 1.03) = P(Z \leq 1.03) - P(Z \leq -1.03) \approx 0.8485 - 0.1515 = 0.6970 \][/tex]
6. Convert to percentage:
- To express this probability as a percentage, multiply by 100:
[tex]\[ 0.6970 \times 100 = 69.7\% \][/tex]
Therefore, approximately 69.7% of outcomes in a standard normal distribution occur within 1.03 standard deviations from the mean.
