Let x be a random variable that represents the length of time it takes a student to complete a take-home exam in Dr. Larson’s psychology class. After interviewing many students, it was found that x has an approximately normal distribution with mean =5.2 and standard deviation =1.8 hours.
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In this scenario, we are given that the random variable x represents the length of time it takes a student to complete a take-home exam in Dr. Larson's psychology class. It is also mentioned that x has an approximately normal distribution with a mean of 5.2 hours and a standard deviation of 1.8 hours.
To find the probability of certain events or values, such as the probability of a student taking less than 4 hours to complete the exam, we can use the information provided about the normal distribution mean and standard deviation.
To calculate probabilities for a normal distribution, we typically use the standard normal distribution table or a statistical software.
To find the probability that a student takes less than 4 hours to complete the exam, we need to find the area under the normal curve to the left of 4, which represents the cumulative probability. Using the standard normal distribution table, we need to calculate the z-score first.
The z-score formula is given by:
z = (x - mean) / standard deviation
For our case:
x = 4 hours
mean = 5.2 hours
standard deviation = 1.8 hours
Substituting the values into the formula:
z = (4 - 5.2) / 1.8
Calculating this:
z = -0.67
Now, we can use the standard normal distribution table to find the cumulative probability corresponding to this z-score. The cumulative probability represents the probability of obtaining a value less than the given z-score.
The standard normal distribution table provides the probabilities for z-scores, usually ranging from 0 to 3. To find the probability of a z-score less than -0.67, we can look up the value in the table.
By looking up the z-score of -0.67 in the standard normal distribution table, we find that the cumulative probability is approximately 0.2514. This means that the probability of a student taking less than 4 hours to complete the exam is approximately 0.2514, or 25.14%.
Note that if you're using statistical software, you can directly input the values of the mean, standard deviation, and specific values to calculate the probabilities without needing to refer to the standard normal distribution table.