During September, the average temperature of Laurel Lake is 64.2º and the standard deviation is 3.2º. Assume the variable is normally distributed. For a randomly selected day, find the probability that the temperature will be
a. Above 62º
b. Below 67º
c. Between 65º and 68º
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
a. .2451
b. .8051
c. .3897
To determine the probability of certain temperatures at Laurel Lake, we can use the Z-score formula and the standard normal distribution.
a. To find the probability that the temperature will be above 62º, we need to calculate the Z-score first. The Z-score is calculated as:
Z = (X - μ) / σ
Where X is the value of interest (62º), μ is the mean (64.2º), and σ is the standard deviation (3.2º).
Using the formula:
Z = (62 - 64.2) / 3.2
Z = -0.688
Now, we can find the probability using the Z-score table or a calculator. The probability above 62º is equal to the probability to the right of the Z-score -0.688.
P(X > 62) = P(Z > -0.688)
By looking up the Z-score in the Z-table, we find that the probability corresponding to -0.688 is approximately 0.752.
Therefore, the probability of the temperature being above 62º is approximately 0.752, or 75.2%.
b. To find the probability that the temperature will be below 67º, we can again calculate the Z-score using the formula:
Z = (X - μ) / σ
Where X is the value of interest (67º), μ is the mean (64.2º), and σ is the standard deviation (3.2º).
Using the formula:
Z = (67 - 64.2) / 3.2
Z = 0.875
Now, we can find the probability using the Z-score table or a calculator. The probability below 67º is equal to the probability to the left of the Z-score 0.875.
P(X < 67) = P(Z < 0.875)
By looking up the Z-score in the Z-table, we find that the probability corresponding to 0.875 is approximately 0.807.
Therefore, the probability of the temperature being below 67º is approximately 0.807, or 80.7%.
c. To find the probability that the temperature will be between 65º and 68º, we need to calculate the Z-scores for the lower and upper bounds of the range.
Lower bound:
Z_lower = (65 - 64.2) / 3.2
Z_lower = 0.250
Upper bound:
Z_upper = (68 - 64.2) / 3.2
Z_upper = 1.187
Now, we can find the probabilities corresponding to these Z-scores using the Z-table.
P(65 < X < 68) = P(0.250 < Z < 1.187)
By looking up the Z-scores in the Z-table, we find that the probabilities corresponding to 0.250 and 1.187 are approximately 0.598 and 0.881, respectively.
Therefore, the probability of the temperature being between 65º and 68º is approximately 0.881 - 0.598 = 0.283, or 28.3%.
To find the probabilities for the given temperature ranges, we can use the standard normal distribution table. However, to do so, we need to first standardize the values using the z-score formula:
z = (x - μ) / σ
where:
- z is the standard score
- x is the raw value
- μ is the mean of the distribution
- σ is the standard deviation
Given:
- Mean (μ) = 64.2º
- Standard deviation (σ) = 3.2º
a. Probability of temperature above 62º:
To find this probability, we need to calculate the area under the normal curve to the right of 62º.
First, let's calculate the z-score using the formula:
z = (x - μ) / σ
z = (62 - 64.2) / 3.2
z = -0.6875
Now, using the z-score, we can find the probability using the standard normal distribution table. The table gives the area to the left of the z-score, so we need to subtract it from 1 to get the area (probability) to the right of the z-score.
Using a standard normal distribution table, the area to the left of -0.6875 is 0.2452. Therefore, the probability of the temperature being above 62º is 1 - 0.2452 = 0.7548, or 75.48%.
b. Probability of temperature below 67º:
Similarly, to find this probability, we need to calculate the area under the normal curve to the left of 67º.
First, let's calculate the z-score using the formula:
z = (x - μ) / σ
z = (67 - 64.2) / 3.2
z = 0.875
Using the z-score, we can find the probability by looking up the area to the left of 0.875 in the standard normal distribution table.
Using the table, the area to the left of 0.875 is 0.8078. Therefore, the probability of the temperature being below 67º is 0.8078, or 80.78%.
c. Probability of temperature between 65º and 68º:
To find this probability, we need to calculate the area under the normal curve between the two values.
First, let's calculate the z-scores for both values:
z1 = (x1 - μ) / σ
z1 = (65 - 64.2) / 3.2
z1 = 0.25
z2 = (x2 - μ) / σ
z2 = (68 - 64.2) / 3.2
z2 = 1.1875
Using the z-scores, we can find the probabilities related to each z-score by looking up the areas to the left of these z-scores in the standard normal distribution table.
Using the table, the area to the left of 0.25 is 0.5987, and the area to the left of 1.1875 is 0.8888.
To find the probability between the two values, we subtract the smaller area from the larger area:
0.8888 - 0.5987 = 0.2901
Therefore, the probability of the temperature being between 65º and 68º is 0.2901, or 29.01%.