Question

normal distributions worksheet
a set of data with a mean of 45 and a standard deviation of
find each value, given its distance from the mean.

1. +1 standard deviation from the mean 2. +3 stand
2. -1 standard deviation from the mean 4. -2 stand

sketch a normal curve for each distribution. label the x-ax
standard deviations from the mean.

1. mean = 95; standard deviation = 12 6. mean =
2. mean = 60; standard deviation = 6 8. mean
3. mea

ition = 60

Explanation:

Looking at the worksheet, the first part has a data set with mean 45, but the standard deviation value is cut off. Let's assume the standard deviation (let's call it \( \sigma \)) was maybe 5 (a common value, or maybe it was a typo, but since the first problem is +1 standard deviation from mean, let's proceed with the formula: value = mean + (z - score * standard deviation). But since the standard deviation is missing, maybe in the original problem it was, say, 5 (common in such worksheets). Wait, maybe the user missed pasting the full standard deviation. Alternatively, maybe it's a typo, but let's check the first problem: "A set of data with a mean of 45 and a standard deviation of [cut off]. Find each value, given its distance from the mean. 1. +1 standard deviation from the mean". Let's assume the standard deviation is, for example, 5 (maybe the original had 5). Then:

Problem 1: +1 standard deviation from mean (mean = 45, let's assume \( \sigma = 5 \) for example, but maybe the actual was 5, 10, etc. Wait, maybe the user's image had a standard deviation, like maybe 5? Wait, no, the image shows "standard deviation of " with the rest cut. Alternatively, maybe it's a mistake, but let's proceed with the formula. The formula for a value at \( z \)-score \( z \) is \( x = \mu + z \cdot \sigma \), where \( \mu \) is mean, \( \sigma \) is standard deviation, \( z \) is the number of standard deviations from mean.

For problem 1: \( z = +1 \), so \( x = 45 + 1 \cdot \sigma \). But since \( \sigma \) is missing, maybe in the original worksheet, the standard deviation was 5 (common). Let's check the next problems. Alternatively, maybe the standard deviation was 5. Let's assume \( \sigma = 5 \) (just for example, but maybe the actual was different). Then:

Step1: Recall the formula for a value in normal distribution: \( x = \mu + z \cdot \sigma \), where \( \mu = 45 \), \( z = +1 \), \( \sigma \) (let's assume \( \sigma = 5 \) as a common value, maybe the original had 5).

\( x = 45 + 1 \times 5 \)

Step2: Calculate the value.

\( x = 45 + 5 = 50 \)

But wait, the standard deviation is missing. Maybe the user made a typo. Alternatively, maybe the standard deviation was 10. Let's check: if \( \sigma = 10 \), then \( x = 45 + 10 = 55 \). But without the correct standard deviation, we can't be sure. Wait, maybe the original worksheet had a standard deviation, like 5, 10, or another number. Alternatively, maybe the user intended to include it. Since the image is partially cut, maybe the standard deviation was 5. Let's proceed with \( \sigma = 5 \) for demonstration.

Problem 3: -1 standard deviation from mean. Using the same \( \mu = 45 \), \( \sigma = 5 \), \( z = -1 \):

Step1: Formula: \( x = \mu + z \cdot \sigma \), \( \mu = 45 \), \( z = -1 \), \( \sigma = 5 \)

\( x = 45 + (-1) \times 5 \)

Step2: Calculate.

\( x = 45 - 5 = 40 \)

But since the standard deviation is missing, this is an assumption. Alternatively, maybe the standard deviation was 10. Let's check: if \( \sigma = 10 \), then problem 1: \( 45 + 10 = 55 \), problem 3: \( 45 - 10 = 35 \).

Alternatively, maybe the user's worksheet had a standard deviation of 5. Let's confirm with the sketch problems. For problem 5: mean = 95, standard deviation = 12. So that's a different problem. So the first part (problems 1-4) has mean 45, standard deviation (let's say) 5 (maybe a typo, but problem 5 has 12). Wait, problem 5: mean = 95, standard deviation = 12. So for problem 5, sketching the normal curve: the normal curve is symmetric around the mean (95). The x-axis labels would be at \( \mu - 3\sigma \), \( \mu - 2\sigma \), \( \mu - \sigma \), \( \mu \), \( \mu + \sigma \), \( \mu + 2\sigma \), \( \mu + 3\sigma \). Let's calculate those:

\( \mu - 3\sigma = 95 - 3 \times 12 = 95 - 36 = 59 \)

\( \mu - 2\sigma = 95 - 24 = 71 \)

\( \mu - \sigma = 95 - 12 = 83 \)

\( \mu = 95 \)

\( \mu + \sigma = 95 + 12 = 107 \)

\( \mu + 2\sigma = 95 + 24 = 119 \)

\( \mu + 3\sigma = 95 + 36 = 131 \)

So the x-axis would be labeled with these values, and the curve is a bell-shaped curve centered at 95, with inflection points at \( \mu \pm \sigma \) (83 and 107).

For problem 7: mean = 60, standard deviation = 6. Similarly, calculate the values:

\( \mu - 3\sigma = 60 - 18 = 42 \)

\( \mu - 2\sigma = 60 - 12 = 48 \)

\( \mu - \sigma = 60 - 6 = 54 \)

\( \mu = 60 \)

\( \mu + \sigma = 60 + 6 = 66 \)

\( \mu + 2\sigma = 60 + 12 = 72 \)

\( \mu + 3\sigma = 60 + 18 = 78 \)

The normal curve is centered at 60, with inflection points at 54 and 66.

But the first part (problems 1-4) has a missing standard deviation. Maybe the original worksheet had a standard deviation, like 5, 10, etc. Let's assume the standard deviation was 5 (common in basic worksheets). Then:

Problem 1: +1 standard deviation from mean

Step1: Use \( x = \mu + z \cdot \sigma \), \( \mu = 45 \), \( z = 1 \), \( \sigma = 5 \)

\( x = 45 + 1 \times 5 \)

Step2: Calculate

\( x = 50 \)

Problem 3: -1 standard deviation from mean

Answer:

Step1: Use \( x = \mu + z \cdot \sigma \), \( \mu = 45 \), \( z = -1 \), \( \sigma = 5 \)

\( x = 45 + (-1) \times 5 \)

Step2: Calculate

\( x = 40 \)

But since the standard deviation is missing, this is an assumption. If the actual standard deviation was different (e.g., 10), the answers would be 55 (problem 1) and 35 (problem 3).

For the sketching problems (5,7), the process is to draw a bell-shaped curve, center it at the mean, label the x-axis with \( \mu - 3\sigma \), \( \mu - 2\sigma \), \( \mu - \sigma \), \( \mu \), \( \mu + \sigma \), \( \mu + 2\sigma \), \( \mu + 3\sigma \), and mark the inflection points at \( \mu \pm \sigma \).

Since the user's question is about solving these, but the standard deviation is missing for problems 1-4, we need to note that. However, assuming the standard deviation was 5 (for example), the answers would be as above. If the actual standard deviation was different, the calculations would change accordingly.

Final Answers (assuming \( \sigma = 5 \) for problems 1-4):

1. \( \boldsymbol{50} \) (if \( \sigma = 5 \))
2. \( \boldsymbol{40} \) (if \( \sigma = 5 \))

For problem 5 (mean=95, \( \sigma=12 \)):

• \( \mu - 3\sigma = 59 \)
• \( \mu - 2\sigma = 71 \)
• \( \mu - \sigma = 83 \)
• \( \mu = 95 \)
• \( \mu + \sigma = 107 \)
• \( \mu + 2\sigma = 119 \)
• \( \mu + 3\sigma = 131 \)

For problem 7 (mean=60, \( \sigma=6 \)):

• \( \mu - 3\sigma = 42 \)
• \( \mu - 2\sigma = 48 \)
• \( \mu - \sigma = 54 \)
• \( \mu = 60 \)
• \( \mu + \sigma = 66 \)
• \( \mu + 2\sigma = 72 \)
• \( \mu + 3\sigma = 78 \)

But since the standard deviation for problems 1-4 is missing, the user should check the original worksheet for the correct standard deviation. If we assume \( \sigma = 5 \), the answers for 1 and 3 are 50 and 40, respectively.