1.The management committee of Zenith has indicated its desire to accept only 40% of all continuing undergraduate based on the results of the end-of-second semester examinations.the mean score of the examination is 500 and standard deviation of 100.Find the cut off score for the exam,assuming the variable normally distributed.

2.Explain why the standard normal distribution can be used to solve many real-life problems.

1. To find the cut-off score for the exam, we can use the concept of z-scores and the standard normal distribution.

Step 1: Convert the desired percentile to a z-score.
The management committee wants to accept only 40% of the students. In order to find the corresponding z-score, you will need to look up the z-table or use a calculator. For a percentile of 40%, the z-score is approximately -0.25.

Step 2: Use the z-score formula to calculate the cut-off score.
The formula for a z-score is: z = (x - μ) / σ
Where:
- z is the z-score,
- x is the observed value,
- μ is the population mean, and
- σ is the population standard deviation.

In this case:
- The z-score is -0.25
- The population mean (μ) is 500
- The population standard deviation (σ) is 100

Rearranging the formula, we have:
x = z * σ + μ
x = (-0.25) * 100 + 500

Simplifying the equation, we get:
x = -25 + 500
x = 475

Therefore, the cut-off score for the exam is 475.

2. The standard normal distribution, also known as the Z-distribution, can be used to solve many real-life problems due to its properties and characteristics. Here are a few reasons:

1. Symmetry: The standard normal distribution is symmetric around its mean, with a bell-shaped curve. This symmetry allows for easier calculations and interpretations of probabilities.

2. Standardization: The standard normal distribution has a mean of zero and a standard deviation of one. This makes it possible to convert any normal distribution into the standard normal distribution by standardizing the values. This simplifies calculations and allows for comparisons across different distributions.

3. Z-scores: The standard normal distribution is used to calculate z-scores, which are measures of how many standard deviations a data point is away from the mean. This allows us to compare values from different data sets and determine the likelihood of an event occurring.

4. Central Limit Theorem: The standard normal distribution is closely related to the concept of the central limit theorem. According to the theorem, the sum or average of a large number of independent and identically distributed random variables will have an approximately normal distribution, regardless of the shape of the original distribution. This property is essential for solving problems involving sample means and proportions.

Overall, the standard normal distribution provides a standard framework for understanding and analyzing data in a wide range of real-life situations, making it a valuable tool in statistics and probability.