Final answer:
To find the probabilities for a standard normal distribution, use the z-table to find the corresponding areas or z-scores. Subtract the area for the lower z from the higher z to find probabilities between two z-scores. For unknown z-scores with given probabilities, identify the area closest to the given probability in the z-table and find the corresponding z-score.
Step-by-step explanation:
To find P(2.51 < z < 2.72), you would typically use a z-table. The table would give you the area to the left of z=2.51 and the area to the left of z=2.72. To find the probability between these two z-scores, you subtract the smaller area from the larger area.
For P(z < c) = 0.4654, you would look for the area closest to 0.4654 in the z-table and find the corresponding z-score. This z-score is the value 'c' that you're looking for.
To find c when P(z > c) = 0.5323, you can use 1 - 0.5323 to find P(z < c), which is 0.4677. Then you would look this up in the z-table and find the corresponding z-score, which is 'c'.
When given a scenario such as thermometer readings or body temperatures, and you have a mean and a standard deviation, you can use the z-score formula z = (X - µ)/σ where X is your value, µ is the mean, and σ is the standard deviation. To find how many standard deviations a score is above the mean, you would plug in the appropriate values. For example, for a data set with a mean of 5 and a standard deviation of 2, and a score of 11, the z-score would be 3, calculated by (11 - 5)/2.
The central limit theorem states that the distribution of sample means will be approximately normal regardless of the shape of the population distribution, provided the sample size is sufficiently large (usually n ≥ 30).