Many body measurements of people of the same sex and similar ages such as height and upper arm length follow a Normal distribution reasonably closely. Weights, on the other hand, are not Normally distributed. Suppose a survey includes the weights of a representative sample of 548548 females in the United States aged 2020 –2929 . Suppose the mean of the weights was 161.55161.55 pounds, and the standard deviation was 48.9748.97 pounds.The histogram of the data includes a smooth curve representing an ?(161.55,48.97)N(161.55,48.97) distribution.From the histogram, the Normal curve does not appear to follow the pattern in the histogram that closely. Because of this, the use of areas under the Normal curve may not provide a good approximation to weights in various intervals.You may find Table A useful.(a) There were 1313 females that weighed under 100100 pounds. What percent of females aged 2020 to 2929 weighed under 100100 pounds? (Enter your answer rounded to two decimal places.)What percent of the ?(161.55,48.97)N(161.55,48.97) distribution is below 100100 ? (Enter your answer rounded to two decimal places.)(b) There were 3333 females that weighed over 250250 pounds. What percent of females aged 2020 to 2929 weighed over 250250 pounds? (Enter your answer rounded to two decimal places.)What percent of the ?(161.55,48.97)N(161.55,48.97) distribution is above 250250 ? (Enter your answer rounded to two decimal places.) ### Understanding Weight DistributionIn this section, we will explore the distribution of weight among a sample population by analyzing a given histogram and examining its distribution curve.#### Histogram AnalysisThe graph presented is a histogram that showcases the distribution of weights within a sample population. The horizontal axis represents the weights in pounds, ranging from 50 to 400. The vertical axis indicates the number of individuals corresponding to each weight range, with values ranging from 0 to 160.#### Key Observations:1. **Weight Ranges**: The histogram is divided into weight intervals, typically in increments of 50 pounds.2. **Peak Frequency**: The highest bar in the histogram is observed in the 150-175 pound range, indicating that the most common weight range within this sample is approximately 150 to 175 pounds. This bar reaches a frequency of nearly 140 individuals.3. **Distribution Shape**: The histogram bars form a bell-shaped distribution, which suggests the data follows a normal distribution pattern. The majority of the sample weights fall between 125 and 225 pounds.4. **Tail Analysis**: The tails of the histogram (left and right extremes) show fewer individuals, indicating that weights below 100 pounds and above 250 pounds are less common in this sample. The right tail extends up to 400 pounds with a very low frequency, showing that very heavy weights are exceptionally rare in this sample.#### Distribution CurveA blue line traces over the histogram bars, portraying the smooth distribution curve of the dataset.- **Bell Curve Representation**: The shape of this line indicates how weights in the sample are normally distributed around the mean. The center of the curve aligns with the highest bar (150-175 pounds), reinforcing the peak frequency observed.- **Symmetry and Spread**: While the curve is relatively symmetrical around the mean, there is a slight skewness to the right, implying more representation of heavier weights in the tail.The analysis of this graph helps us understand the central tendency and the spread of the weights within the sample population. By exploring distributions like this, we can gain insights into the typical weight ranges and identify outliers in a population.---This histogram serves as an excellent educational tool for illustrating the concept of normal distributions and statistical analysis of real-world data. **Statistical Analysis of Female Weight Distribution**This exercise involves analyzing the distribution of weights for females aged 20 to 29. We compare actual weight percentages with predicted percentages using the normal distribution, $N(161.55, 48.97^2)$.---### Part (a)#### Problem:There were 13 females that weighed under 100 pounds. What percent of females aged 20 to 29 weighed under 100 pounds? (Enter your answer rounded to two decimal places.)**Calculation:**percent under 100 lbs: \[ 2.19\% \]**Feedback:** Incorrect#### Follow-Up Question:What percent of the $N(161.55, 48.97)$ distribution is below 100? (Enter your answer rounded to two decimal places.)**Calculation:**percent below 100:\[ \text{(Your answer here)}\% \]**Feedback:** Incorrect---### Part (b)#### Problem:There were 33 females that weighed over 250 pounds. What percent of females aged 20 to 29 weighed over 250 pounds? (Enter your answer rounded to two decimal places.)**Calculation:**percent over 250 lbs:\[ 6.20 \% \]**Feedback:** Incorrect#### Follow-Up Question:What percent of the $N(161.55, 48.97)$ distribution is above 250? (Enter your answer rounded to two decimal places.)**Calculation:**percent above 250:\[ \text{(Your answer here)}\% \]**Feedback:** Incorrect---### Part (c)#### Discussion:Based on your answers in parts (a) and (b), do you think it is a good idea to summarize the distribution of weights by an $N(161.55, 48.97)$ distribution?- Yes, because the predicted percentages are not significantly different from the actual percentages.- No, because the predicted percentages are significantly different than the actual percentages.- Yes, because most females weigh between 100 pounds and 250 pounds.- No, because the sample size is too small.**Feedback:** Incorrect---**Explanation of Statistical Tools:**- **Normal Distribution $N(161.55, 48.97^2)$:** This represents a normal distribution where the mean weight is 161.55 pounds, and the variance is calculated as the square of 48.97

Given , according the question , a survey includes the weight of a representative samples 548…

Many body measurements of people of the same sex and similar ages such as height and upper arm length follow a Normal distribution reasonably closely. Weights, on the other hand, are not Normally distributed. Suppose a survey includes the weights of a representative sample of 548548 females in the United States aged 2020 –2929 . Suppose the mean of the weights was 161.55161.55 pounds, and the standard deviation was 48.9748.97 pounds. The histogram of the data includes a smooth curve representing an ?(161.55,48.97)N(161.55,48.97) distribution. From the histogram, the Normal curve does not appear to follow the pattern in the histogram that closely. Because of this, the use of areas under the Normal curve may not provide a good approximation to weights in various intervals. You may find Table A useful. (a) There were 1313 females that weighed under 100100 pounds. What percent of females aged 2020 to 2929 weighed under 100100 pounds? (Enter your answer rounded to two decimal places.) What percent of the ?(161.55,48.97)N(161.55,48.97) distribution is below 100100 ? (Enter your answer rounded to two decimal places.) (b) There were 3333 females that weighed over 250250 pounds. What percent of females aged 2020 to 2929 weighed over 250250 pounds? (Enter your answer rounded to two decimal places.) What percent of the ?(161.55,48.97)N(161.55,48.97) distribution is above 250250 ? (Enter your answer rounded to two decimal places.)

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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Many body measurements of people of the same sex and similar ages such as height and upper arm length follow a Normal distribution reasonably closely. Weights, on the other hand, are not Normally distributed. Suppose a survey includes the weights of a representative sample of 548548 females in the United States aged 2020 –2929 . Suppose the mean of the weights was 161.55161.55 pounds, and the standard deviation was 48.9748.97 pounds.

The histogram of the data includes a smooth curve representing an ?(161.55,48.97)N(161.55,48.97) distribution.

From the histogram, the Normal curve does not appear to follow the pattern in the histogram that closely. Because of this, the use of areas under the Normal curve may not provide a good approximation to weights in various intervals.

You may find Table A useful.

(a) There were 1313 females that weighed under 100100 pounds. What percent of females aged 2020 to 2929 weighed under 100100 pounds? (Enter your answer rounded to two decimal places.)

What percent of the ?(161.55,48.97)N(161.55,48.97) distribution is below 100100 ? (Enter your answer rounded to two decimal places.)

(b) There were 3333 females that weighed over 250250 pounds. What percent of females aged 2020 to 2929 weighed over 250250 pounds? (Enter your answer rounded to two decimal places.)

What percent of the ?(161.55,48.97)N(161.55,48.97) distribution is above 250250 ? (Enter your answer rounded to two decimal places.)

### Understanding Weight Distribution

In this section, we will explore the distribution of weight among a sample population by analyzing a given histogram and examining its distribution curve.

#### Histogram Analysis

The graph presented is a histogram that showcases the distribution of weights within a sample population. The horizontal axis represents the weights in pounds, ranging from 50 to 400. The vertical axis indicates the number of individuals corresponding to each weight range, with values ranging from 0 to 160.

#### Key Observations:

1. **Weight Ranges**: The histogram is divided into weight intervals, typically in increments of 50 pounds.
2. **Peak Frequency**: The highest bar in the histogram is observed in the 150-175 pound range, indicating that the most common weight range within this sample is approximately 150 to 175 pounds. This bar reaches a frequency of nearly 140 individuals.
3. **Distribution Shape**: The histogram bars form a bell-shaped distribution, which suggests the data follows a normal distribution pattern. The majority of the sample weights fall between 125 and 225 pounds.
4. **Tail Analysis**: The tails of the histogram (left and right extremes) show fewer individuals, indicating that weights below 100 pounds and above 250 pounds are less common in this sample. The right tail extends up to 400 pounds with a very low frequency, showing that very heavy weights are exceptionally rare in this sample.

#### Distribution Curve

A blue line traces over the histogram bars, portraying the smooth distribution curve of the dataset.

- **Bell Curve Representation**: The shape of this line indicates how weights in the sample are normally distributed around the mean. The center of the curve aligns with the highest bar (150-175 pounds), reinforcing the peak frequency observed.
- **Symmetry and Spread**: While the curve is relatively symmetrical around the mean, there is a slight skewness to the right, implying more representation of heavier weights in the tail.

The analysis of this graph helps us understand the central tendency and the spread of the weights within the sample population. By exploring distributions like this, we can gain insights into the typical weight ranges and identify outliers in a population.

---

This histogram serves as an excellent educational tool for illustrating the concept of normal distributions and statistical analysis of real-world data.

**Statistical Analysis of Female Weight Distribution**

This exercise involves analyzing the distribution of weights for females aged 20 to 29. We compare actual weight percentages with predicted percentages using the normal distribution, $N(161.55, 48.97^2)$.

---

### Part (a)

#### Problem:

There were 13 females that weighed under 100 pounds. What percent of females aged 20 to 29 weighed under 100 pounds? (Enter your answer rounded to two decimal places.)

**Calculation:**

percent under 100 lbs:
\[ 2.19\% \]
**Feedback:** Incorrect

#### Follow-Up Question:

What percent of the $N(161.55, 48.97)$ distribution is below 100? (Enter your answer rounded to two decimal places.)

**Calculation:**

percent below 100:
\[ \text{(Your answer here)}\% \]
**Feedback:** Incorrect

---

### Part (b)

#### Problem:

There were 33 females that weighed over 250 pounds. What percent of females aged 20 to 29 weighed over 250 pounds? (Enter your answer rounded to two decimal places.)

**Calculation:**

percent over 250 lbs:
\[ 6.20 \% \]
**Feedback:** Incorrect

#### Follow-Up Question:

What percent of the $N(161.55, 48.97)$ distribution is above 250? (Enter your answer rounded to two decimal places.)

**Calculation:**

percent above 250:
\[ \text{(Your answer here)}\% \]
**Feedback:** Incorrect

---

### Part (c)

#### Discussion:

Based on your answers in parts (a) and (b), do you think it is a good idea to summarize the distribution of weights by an $N(161.55, 48.97)$ distribution?

- Yes, because the predicted percentages are not significantly different from the actual percentages.
- No, because the predicted percentages are significantly different than the actual percentages.
- Yes, because most females weigh between 100 pounds and 250 pounds.
- No, because the sample size is too small.

**Feedback:** Incorrect

---

**Explanation of Statistical Tools:**

- **Normal Distribution $N(161.55, 48.97^2)$:** This represents a normal distribution where the mean weight is 161.55 pounds, and the variance is calculated as the square of 48.97

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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