Normal distribution: probability below a score

1. Audio
   Observation

   Mean of the scores

2. Diagram
   Observation

   The number 70 is labeled in the center of the diagram

$\mu$

Mean of the scores

Real numbers

1. Audio
   Observation

   Standard deviation of the scores

2. Diagram
   Observation

   Two 10s are labeled below the diagram, on the left and right sides of 70 respectively

$\sigma$

Standard deviation of the scores

Positive real numbers

1. Audio
   Observation

   The graph of a normal distribution is a symmetric bell-shaped curve, whose highest point corresponds to the mean.

2. Diagram
   Observation

   A bell-shaped curve appears on the screen

The graph of a normal distribution is a symmetric bell-shaped curve, whose highest point corresponds to the mean.

1. Data follows a normal distribution

1. Audio
   Observation

   Using the68-95 approximation, the diagram labels the two central halves34% each and the tails16% each.

2. Diagram
   Observation

   Percentage regions of 34 and 16 are labeled on the diagram

In a normal distribution, the probability that data falls within one standard deviation of the mean is approximately 68% (34% on each side); the probability that it falls within two standard deviations is approximately 95% (the remaining two tails each account for about 2.5%, but this problem uses 16% as the approximate value for one tail beyond one standard deviation, i.e., (100%-68%)/2 = 16%).

1. Data follows a normal distribution

2. Use rounded probabilities under the stated continuous normal model; the95%/2.5% figures describe the broader rule, not an additional calculation in this exercise.

1. Normal Distribution Graph

1. Audio
   Observation

   The solution identifies60 as one standard deviation below70 and chooses the approximate left-tail option0.16.

2. Diagram
   Observation

   The region less than 60 points on the left side of the diagram is circled and labeled 16%

1. Expression

   $$\mu = 70, \sigma = 10$$
   Explanation

   Determine the mean and standard deviation of the normal distribution

   Justification

   Given conditions in the problem

   Shown in the video
2. Expression

   $$60 = \mu - \sigma$$
   Explanation

   Calculate the standard deviation score corresponding to 60 points

   Justification

   Algebraic operation

   Derived from the video
3. Expression

   $$P(X < 60) = P(X < \mu - \sigma)$$
   Explanation

   Find the probability of a score below the mean minus one standard deviation.

   Justification

   Substitution of equivalent quantities

   Supplementary explanation
4. Expression

   $$P(X < \mu - \sigma) \approx 16\% = 0.16$$
   Explanation

   Use the approximate68-95 rule to choose the closest option; the tail probability is not exactly16%.

   Justification

   Properties of normal distribution

   Shown in the video

The probability that the score is lower than 60 points is closest to 0.16.

1. Caption evidence
   Observation

   The question specifies normal scores with mean70 and standard deviation10, asks for the probability below60, and lists option1 as0.16.

2. Audio
   Observation

   The speaker explains this problem throughout

It is known that student scores follow a normal distribution, with a mean of 70 and a standard deviation of 10. Find the probability that a randomly selected student has a score lower than 60.

1. Scores follow a normal distribution

2. Mean $\mu = 70$

3. Standard deviation $\sigma = 10$

4. Target score x = 60

Find P(X < 60)

1. Expression

   $$\mu = 70, \sigma = 10$$
   Explanation

   Mark the positions of the mean and standard deviation

   Justification

   Given by the problem

   Shown in the video
2. Expression

   $$60 = 70 - 10 = \mu - \sigma$$
   Explanation

   Discover that 60 points is exactly the mean minus one standard deviation

   Justification

   Algebraic calculation

   Derived from the video
3. Expression

   $$P(X < \mu - \sigma) \approx 16\%$$
   Explanation

   According to the 68-95 rule, the probability of the single-sided tail is approximately 16%

   Justification

   Empirical rule of normal distribution

   Shown in the video
4. Expression

   $$16\% = 0.16$$
   Explanation

   Convert percentage to decimal

   Justification

   Unit conversion

   Shown in the video

(1) 0.16

Comparing with the options, (1) is 0.16, which matches the calculation result.

1. Animation
   Observation

   Gradually drawing the normal distribution curve, labeling values such as 70, 10, 34%, 16%, and circling the region less than 60 points

1. Bell-shaped curve

2. Vertical dashed line

3. Horizontal axis

4. Percentage values

5. Circled region

1. Draw the curve

2. Label the mean 70

3. Label the standard deviation 10

4. Label interval probabilities 34% and 16%

5. Circle the target region

1. Curve shape

2. Symmetry

Visually presents the parameters and probability distribution of the normal distribution, helping to understand the relative position of 60 points and its corresponding probability.

1. Audio
   Observation

   The diagram and empirical rule identify the left tail below60 as approximately16%.

The 68-95 rule is the theoretical basis for solving this probability calculation problem.

1. Audio
   Observation

   How to use the 68-95 rule of normal distribution to calculate the probability below a specific score?

1. 68-95 Rule
2. Derivation of Probability Less Than 60 Points
3. Taiwan Math B entrance exam, ROC year99 single-choice question2