Question

assume that thermometer readings are normally distributed with a mean of 0°c and a standard deviation of 1.00°c. a thermometer is randomly selected and tested. for the case below, draw a sketch, and find the probability of the reading. (the given values are in celsius degrees.) between 1.00 and 2.25. click to view page 1 of the table. click to view page 2 of the table. draw a sketch. choose the correct graph below. (options a, b, c with images: a, b, c show normal curves with z=1.00 and z=2.25, different shaded regions.)

Explanation:

Part 1: Choosing the Correct Graph

To determine the correct sketch for the probability between \( z = 1.00 \) and \( z = 2.25 \) in a standard normal distribution:

• The standard normal curve is symmetric around \( z = 0 \), with the peak at \( z = 0 \).
• We want the area between \( z = 1.00 \) and \( z = 2.25 \), which is the region to the right of \( z = 1.00 \) and to the left of \( z = 2.25 \) (since both \( z \)-scores are positive, they lie to the right of the mean \( z = 0 \)).

Now, analyze the options:

• Option A: Shows a small shaded area between \( z = 1.00 \) and \( z = 2.25 \), which matches the region between two positive \( z \)-scores.
• Option B: The shaded area is only at \( z = 2.25 \) (extreme right), not between \( z = 1.00 \) and \( z = 2.25 \).
• Option C: The shaded area is to the left of \( z = 1.00 \) (or a larger region), which does not represent the area between \( z = 1.00 \) and \( z = 2.25 \).

Thus, the correct graph is Option A.

Part 2: Calculating the Probability

To find \( P(1.00 < Z < 2.25) \) for a standard normal variable \( Z \):
Recall that for a standard normal distribution, \( P(a < Z < b) = P(Z < b) - P(Z < a) \), where \( P(Z < z) \) is the cumulative probability from the left up to \( z \).

Step 1: Find \( P(Z < 2.25) \)

Using the standard normal table (z-table):

• For \( z = 2.25 \), the cumulative probability \( P(Z < 2.25) = 0.9878 \) (from z-table: row 2.2, column 0.05).

Step 2: Find \( P(Z < 1.00) \)

Using the standard normal table:

• For \( z = 1.00 \), the cumulative probability \( P(Z < 1.00) = 0.8413 \) (from z-table: row 1.0, column 0.00).

Step 3: Compute the Difference

\[
P(1.00 < Z < 2.25) = P(Z < 2.25) - P(Z < 1.00) = 0.9878 - 0.8413 = 0.1465
\]

Final Answers

• Correct graph: \(\boldsymbol{\text{Option A}}\)
• Probability: \(\boldsymbol{0.1465}\) (or 14.65%)

Answer:

Part 1: Choosing the Correct Graph

To determine the correct sketch for the probability between \( z = 1.00 \) and \( z = 2.25 \) in a standard normal distribution:

• The standard normal curve is symmetric around \( z = 0 \), with the peak at \( z = 0 \).
• We want the area between \( z = 1.00 \) and \( z = 2.25 \), which is the region to the right of \( z = 1.00 \) and to the left of \( z = 2.25 \) (since both \( z \)-scores are positive, they lie to the right of the mean \( z = 0 \)).

Now, analyze the options:

• Option A: Shows a small shaded area between \( z = 1.00 \) and \( z = 2.25 \), which matches the region between two positive \( z \)-scores.
• Option B: The shaded area is only at \( z = 2.25 \) (extreme right), not between \( z = 1.00 \) and \( z = 2.25 \).
• Option C: The shaded area is to the left of \( z = 1.00 \) (or a larger region), which does not represent the area between \( z = 1.00 \) and \( z = 2.25 \).

Thus, the correct graph is Option A.

Part 2: Calculating the Probability

To find \( P(1.00 < Z < 2.25) \) for a standard normal variable \( Z \):
Recall that for a standard normal distribution, \( P(a < Z < b) = P(Z < b) - P(Z < a) \), where \( P(Z < z) \) is the cumulative probability from the left up to \( z \).

Step 1: Find \( P(Z < 2.25) \)

Using the standard normal table (z-table):

• For \( z = 2.25 \), the cumulative probability \( P(Z < 2.25) = 0.9878 \) (from z-table: row 2.2, column 0.05).

Step 2: Find \( P(Z < 1.00) \)

Using the standard normal table:

• For \( z = 1.00 \), the cumulative probability \( P(Z < 1.00) = 0.8413 \) (from z-table: row 1.0, column 0.00).

Step 3: Compute the Difference

\[
P(1.00 < Z < 2.25) = P(Z < 2.25) - P(Z < 1.00) = 0.9878 - 0.8413 = 0.1465
\]

Final Answers

• Correct graph: \(\boldsymbol{\text{Option A}}\)
• Probability: \(\boldsymbol{0.1465}\) (or 14.65%)