(a) Find the area under the standard normal curve to the right of z=0.78.

To find the area under the standard normal curve to the right of \( z = 0.78 \), we can use the standard normal distribution table (Z-table) or a calculator.

1. Look up the Z-score: The Z-table provides the area to the left of a given Z-score. For \( z = 0.78 \), the area to the left is approximately \( 0.7823 \).

2. Calculate the area to the right: Since the total area under the normal curve is 1, to find the area to the right, we subtract the area to the left from 1: \[ \text{Area to the right} = 1 - \text{Area to the left} \] \[ \text{Area to the right} = 1 - 0.7823 = 0.2177 \]

Thus, the area to the right of \( z = 0.78 \) is approximately 0.2177.

b) Find the area under the standard normal curve to the right of z=-1.81.

To find the area under the standard normal curve to the right of \( z = -1.81 \), you can follow a similar process:

1. Look up the Z-score: Check the Z-table for the area to the left of \( z = -1.81 \). The area to the left (cumulative probability) is approximately \( 0.0351 \).

2. Calculate the area to the right: To find the area to the right, subtract the area to the left from 1: \[ \text{Area to the right} = 1 - \text{Area to the left} \] \[ \text{Area to the right} = 1 - 0.0351 = 0.9649 \]

Thus, the area to the right of \( z = -1.81 \) is approximately 0.9649.

(a) Find the area under the standard normal curve that lies between -1.71 and 1.25.

To find the area under the standard normal curve that lies between \( z = -1.71 \) and \( z = 1.25 \), you can follow these steps:

1. Find the area to the left of \( z = 1.25 \):

   • Using a Z-table or calculator, the area to the left of \( z = 1.25 \) is approximately \( 0.8944 \).

2. Find the area to the left of \( z = -1.71 \):

   • The area to the left of \( z = -1.71 \) is approximately \( 0.0436 \).

3. Calculate the area between \( z = -1.71 \) and \( z = 1.25 \):

   • To find the area between these two Z-scores, subtract the area to the left of \( z = -1.71 \) from the area to the left of \( z = 1.25 \): \[ \text{Area between} = P(Z < 1.25) - P(Z < -1.71) \] \[ \text{Area between} = 0.8944 - 0.0436 = 0.8508 \]

Therefore, the area under the standard normal curve that lies between \( z = -1.71 \) and \( z = 1.25 \) is approximately 0.8508.

(b) Find the area under the standard normal curve that lies between -1.92 and -1.17.

To find the area under the standard normal curve that lies between \( z = -1.92 \) and \( z = -1.17 \), follow these steps:

1. Find the area to the left of \( z = -1.17 \):

   • Using a Z-table or calculator, the area to the left of \( z = -1.17 \) is approximately \( 0.1210 \).

2. Find the area to the left of \( z = -1.92 \):

   • The area to the left of \( z = -1.92 \) is approximately \( 0.0274 \).

3. Calculate the area between \( z = -1.92 \) and \( z = -1.17 \):

   • To find the area between these two Z-scores, subtract the area to the left of \( z = -1.92 \) from the area to the left of \( z = -1.17 \): \[ \text{Area between} = P(Z < -1.17) - P(Z < -1.92) \] \[ \text{Area between} = 0.1210 - 0.0274 = 0.0936 \]

Therefore, the area under the standard normal curve that lies between \( z = -1.92 \) and \( z = -1.17 \) is approximately 0.0936.

Use the TI-84 Plus calculator to find the z-score for which the area to its left is 0.59. Round the answer to two decimal places.

The z-score for the given area is ?

To find the z-score corresponding to an area of \( 0.59 \) to its left using a TI-84 Plus calculator, you can follow these steps:

1. Turn on the calculator and press the 2nd button.
2. Access the DISTR menu by pressing the VARS key.
3. Select the inverse normal function:
   • Scroll to invNorm( or just press 2.
4. Input the area and the parameters for the standard normal distribution:
   • Type invNorm(0.59).
5. Close the parentheses and press ENTER.

The TI-84 Plus will return the z-score corresponding to an area of \( 0.59 \).

After performing these steps, you should find that the z-score is approximately 0.28 when rounded to two decimal places.

Thus, the z-score for which the area to its left is \( 0.59 \) is 0.28.

Use the TI-84 Plus calculator to find the z-scores that bound the middle 88% of the area under the standard normal curve. Enter the answers in ascending order and round to two decimal places.

The z-scores for the given area are [?] and [?]

To find the z-scores that bound the middle 88% of the area under the standard normal curve using a TI-84 Plus calculator, you can follow these steps:

1. Calculate the area in each tail: Since you want to find the z-scores that bound the middle 88% of the area, the remaining area in the tails is \( 100% - 88% = 12% \). Since there are two tails, you divide this by 2: \[ \text{Area in each tail} = \frac{12%}{2} = 6% = 0.06 \]

2. Find the area to the left of the lower z-score: This is the area to the left of the lower z-score: \[ \text{Area to the left of lower z-score} = 0.06 \]

3. Find the area to the left of the upper z-score: This is the area to the left of the upper z-score: \[ \text{Area to the left of upper z-score} = 1 - 0.06 = 0.94 \]

4. Use invNorm to find the z-scores:

   • For the lower z-score:
     • Press 2nd then VARS to access the DISTR menu.
     • Select invNorm( and enter 0.06: \[ \text{Lower z-score} = \text{invNorm}(0.06) \]
   • For the upper z-score:
     • Again, go to the DISTR menu.
     • Select invNorm( and enter 0.94: \[ \text{Upper z-score} = \text{invNorm}(0.94) \]

5. Calculate the z-scores:

   • After pressing ENTER for both, you will find:
     • Lower z-score \( \approx -1.55 \)
     • Upper z-score \( \approx 1.55 \)

Thus, the z-scores that bound the middle 88% of the area under the standard normal curve are -1.55 and 1.55 (rounded to two decimal places).

So your final answer is:

• Lower z-score: -1.55
• Upper z-score: 1.55