A gallup poll of 1236 adults showed that 14% believe that bad luck follows if your path is crossed by a black cat, so n=1236 and p=0.14
1.Find the mean?
2.Find the standard deviation?
3.Find the ND (np > 5 and nq > 5 ?)
4.Find the min, max of usual values by RROT
1. Mean = 1236 * 0.14 = 173.04
2. Standard deviation = √(1236 * 0.14 * 0.86) = 128.9077
3. ND = 1236 * 0.14 * 0.86 = 106.7144
4. Min = 0, Max = 1236
1. To find the mean (μ), multiply the sample size (n) by the observed proportion (p):
μ = n * p = 1236 * 0.14 = 172.44 (rounded to 2 decimal places).
Therefore, the mean is approximately 172.44.
2. To find the standard deviation (σ), first calculate the standard error (SE) using the formula:
SE = sqrt(n * p * q)
where q is equal to 1 - p.
SE = sqrt(1236 * 0.14 * 0.86) = 7.585 (rounded to 3 decimal places).
Next, calculate the standard deviation by multiplying the standard error by the square root of the sample size:
σ = SE * sqrt(n) = 7.585 * sqrt(1236) = 7.585 * 35.128 = 266.846 (rounded to 3 decimal places).
Therefore, the standard deviation is approximately 266.846.
3. To determine if the normal distribution assumption is valid, we need to check if both np and nq are greater than 5, where np represents the expected number of successes and nq represents the expected number of failures.
np = 1236 * 0.14 = 172.44 (rounded to 2 decimal places).
nq = 1236 * 0.86 = 1063.76 (rounded to 2 decimal places).
Since both np and nq are greater than 5 (172.44 > 5 and 1063.76 > 5), the normal distribution assumption is valid.
4. The usual values of a normal distribution can be determined using the rule of thumb known as the "empirical rule" or "68-95-99.7 rule", which states that:
- Approximately 68% of the values will fall within one standard deviation of the mean.
- Approximately 95% of the values will fall within two standard deviations of the mean.
- Approximately 99.7% of the values will fall within three standard deviations of the mean.
Min usual value = mean - (1 * standard deviation) = 172.44 - (1 * 266.846) = -94.406 (rounded to 3 decimal places).
Max usual value = mean + (1 * standard deviation) = 172.44 + (1 * 266.846) = 439.686 (rounded to 3 decimal places).
Therefore, the usual range of values is approximately -94.406 to 439.686.
To find the mean, standard deviation, ND, and the minimum and maximum of usual values, we can start by understanding the formulas and concepts used.
1. Mean:
The mean (μ) is the expected value or average for a distribution and is calculated by multiplying the sample size (n) by the probability of success (p): μ = np. In this case, μ = 1236 * 0.14 = 172.44 (approximately).
2. Standard Deviation:
The standard deviation (σ) measures the amount of variability or dispersion in the data. For a binomial distribution, the standard deviation is given by the formula: σ = √(npq), where q = 1 - p. Thus, the standard deviation in this case would be: σ = √(1236 * 0.14 * (1 - 0.14)) = √(172.3864) ≈ 13.12.
3. ND (np > 5 and nq > 5):
To determine if the normal distribution approximation applies (nd), we need np and nq to both be greater than 5. In this case, np = 1236 * 0.14 = 172.44, and nq = 1236 * 0.86 = 1061.76. Since both np and nq are greater than 5, the normal distribution approximation applies.
4. Min and Max of Usual Values by RROT:
The "min and max of usual values by RROT" refers to calculating the range of values that would contain the usual or expected outcomes within a certain range.
To find this range, we can use the "Rule of Thumb" or the "Empirical Rule" which states that for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
So, to find the minimum and maximum of usual values by RROT, we can calculate as follows:
Minimum Usual Value = Mean - (3 * Standard Deviation)
= 172.44 - (3 * 13.12)
≈ 133.08
Maximum Usual Value = Mean + (3 * Standard Deviation)
= 172.44 + (3 * 13.12)
≈ 211.8
Therefore, the usual range of values within 3 standard deviations of the mean is approximately 133.08 to 211.8.