from a set of 1000 observations known to be normally distributed the mean is 534 cm and sd is 13.5 cm. how many observations are likely to exceed 561 cm, how many will be between 520.5 cm 547.5 cm, between what limits will the middle 50% of the observations lie

To answer these questions, we'll use the properties of the normal distribution and its associated z-scores. The z-score represents the number of standard deviations an observation is away from the mean.

1. How many observations are likely to exceed 561 cm?
To find the number of observations likely to exceed a certain value, we need to calculate the area under the normal curve to the right of that value.

Step 1: Calculate the z-score of 561 cm.
z = (561 - mean) / standard deviation
z = (561 - 534) / 13.5
z ≈ 2

Step 2: Determine the area to the right of the z-score using a standard normal distribution table or calculator. The area to the right of z = 2 is approximately 0.0228.

Step 3: Multiply the area by the total number of observations:
Number of observations likely to exceed 561 cm = 0.0228 * 1000 ≈ 22.8 (~23 observations)

Therefore, approximately 23 observations are likely to exceed 561 cm.

2. How many observations will be between 520.5 cm and 547.5 cm?
Similar to the first question, we need to find the area under the normal curve between two given values.

Step 1: Calculate the z-scores of 520.5 cm and 547.5 cm.
z1 = (520.5 - mean) / standard deviation
z1 = (520.5 - 534) / 13.5 ≈ -0.963

z2 = (547.5 - mean) / standard deviation
z2 = (547.5 - 534) / 13.5 ≈ 1.000

Step 2: Determine the area between z1 and z2 using a standard normal distribution table or calculator. The area between z = -0.963 and z = 1.000 is approximately 0.6725.

Step 3: Multiply the area by the total number of observations:
Number of observations between 520.5 cm and 547.5 cm = 0.6725 * 1000 ≈ 672.5 (~673 observations)

Therefore, approximately 673 observations will be between 520.5 cm and 547.5 cm.

3. Between what limits will the middle 50% of the observations lie?
The middle 50% of the observations corresponds to the range between the 25th percentile and the 75th percentile of the distribution.

Step 1: Convert the percentiles to z-scores. The z-scores corresponding to the 25th and 75th percentiles are -0.674 and 0.674, respectively.

Step 2: Convert the z-scores back to actual values using the mean and standard deviation.
Lower limit = mean + (z-score * standard deviation)
Lower limit = 534 + (-0.674 * 13.5) ≈ 524.19 cm

Upper limit = mean + (z-score * standard deviation)
Upper limit = 534 + (0.674 * 13.5) ≈ 543.81 cm

Therefore, the middle 50% of the observations will lie between approximately 524.19 cm and 543.81 cm.