Normal Probabilityenter Input In Blue Cells See Answers In Yellow Cel ✓ Solved

Normal probability Enter input in blue cells; See answers in yellow cells ANSWER Z score x 3.5 Z score -2 mean 6.5 standard deviation 1.5 LESS THAN(At most) AREA LESS THAN ENTER Z score 0.8 Asked to find x 32 Probability Less than F7 0.7881 Mean 28 Stdev 5 MORE THAN (at least) AREA MORE THAN ENTER Z score -0. Asked to find x 12 Probability more than F13 0.7475 Mean 14 Stdev 3 Probability INBETWEEN ENTER ANSWER INBETWEEN Asked to find Large value 370 Probability between F19 and F20 is 0.6904 Asked to find smaller value 360 Mean 365.45 Stdev 4.9 Find x given probability Enter input in blue cells; See answers in yellow cells Find x given Area to Right or Left Enter Area to the left no If area to the left is given enter yes; otherwise say no Area 0.03 enter decimal SUMMARY Enter Mean 47500 z 1.

Enter Standard deviation 3000 Find x 53142. Find X given Area in the middle Area 0.5 enter decimal SUMMARY Enter Mean 81 Lower x 78. Enter Standard deviation 4 Upper x 83. Empirical Rule Enter input in blue cells; See answers in yellow cells ANSWER Empirical Rule .7 mean 92 Lower number Upper number standard deviation % % .70% CLT Theorem Enter input in blue cells; See answers in yellow cells CLT THEOREM FOR MEANS Standard error 2.8366 n 10 Standard deviation 8.97 CLT THEOREM FOR PROPORTION Standard error 0.0913 n 30 proportion of success 0.5 Pearson work. Here is the link of sign in pearson website.

Go to Math 1B and take chapter 8 assignments and the second attempt of test 1. List here: hw 8.5, 8.6, 8.8, 8.9 and Quiz 4 (sec 8.5, 8.6, 8.9) It’s due on end of April 2nd. Ask me for the account and password. Descriptives Put values in blue cells; output or answers in YELLOW cells Data 1 Mean 4. Median 4.

Mode 1. ERROR:#N/A ERROR:#N/A ERROR:#N/A (Returns more than one mode) 2 Sample Variance 12. Sample Standard Deviation 3. Population Variance 11. Population Standard Deviation 3.

Range 10. Count (n) 9.0000 Min 1.0000 Quartile 1 1.5000 Median 4.0000 Quartile3 7.5000 Max 11.0000 Interquartile Range (IQR) 6.0000

Paper for above instructions

Overview of Normal Probability Distributions, Empirical Rule, Central Limit Theorem, and Descriptive Statistics

In the realm of statistics, understanding normal probability distributions and related concepts such as the Empirical Rule and the Central Limit Theorem (CLT) is essential for data analysis. This paper delves into interpreting data inputs related to normal probability, calculating Z-scores, and applying descriptive statistics. The following sections detail these concepts and apply them to the provided data, ensuring a comprehensive understanding is conveyed.

Normal Probability Distributions

Normal Distribution is a continuous probability distribution that is symmetrical and characterized by its bell-shaped curve. It is defined by two parameters: the mean (μ) and the standard deviation (σ). The standard normal distribution is a special case where the mean μ = 0 and the standard deviation σ = 1. For any random variable \( X \) following a normal distribution, the Z-score is computed as:
\[ Z = \frac{(X - \mu)}{\sigma} \]
where \( Z \) represents the Z-score, \( X \) is the value of interest, μ is the mean, and σ is the standard deviation.

Calculations Based on Provided Data

1. Z-Score and Probability less than a specified value

For the input data:
- Z-score for \( x = 3.5\) where \( \mu = 6.5\), \( \sigma = 1.5 \):
\[ Z = \frac{(3.5 - 6.5)}{1.5} = \frac{-3}{1.5} = -2 \]
The probability of a random variable being less than \( 3.5 \) can be found using Z-tables or normal distribution calculators. The cumulative probability associated with a Z-score of -2 is approximately 0.0228 (Z-Score Lookup Table).

2. Finding x given a probability

For a probability of less than 0.7881 and mean 28 with a standard deviation of 5:
Using the Z-table, we find that Z corresponds to 0.7881 is approximately 0.77. Therefore,
\[ x = \mu + Z \cdot \sigma = 28 + (0.77 \cdot 5) = 28 + 3.85 = 31.85 \]
Rounding gives \( x \approx 32 \).

3. Probability of more than a specified value

For \( P(X > 12) \) where the mean is 14 and standard deviation is 3:
Calculate the Z-score:
\[ Z = \frac{(12 - 14)}{3} = \frac{-2}{3} \approx -0.67 \]
Using the Z-table, \( P(Z < -0.67) \approx 0.2514 \), hence:
\[ P(X > 12) = 1 - P(Z < -0.67) = 1 - 0.2514 = 0.7486 \]

4. Probability between two values

Between \( x = 360 \) and \( x = 370 \) where mean is 365.45 and standard deviation is 4.9:
Calculating Z-scores:
- For \( x = 360 \):
\[ Z_{360} = \frac{(360 - 365.45)}{4.9} \approx -1.11 \]
- For \( x = 370 \):
\[ Z_{370} = \frac{(370 - 365.45)}{4.9} \approx 0.11 \]
Using Z-tables:
- \( P(Z < -1.11) \approx 0.1335\)
- \( P(Z < 0.11) \approx 0.5438\)
Therefore, the probability between 360 and 370 is:
\[ P(360 < X < 370) = P(Z < 0.11) - P(Z < -1.11) = 0.5438 - 0.1335 = 0.4103 \]

The Empirical Rule

The Empirical Rule states that for a normal distribution:
- Approximately 68% of data falls within one standard deviation from the mean.
- About 95% falls within two standard deviations.
- Roughly 99.7% is within three standard deviations.
Using these concepts, if we take a mean of 92 and a standard deviation of 7, applying the Empirical Rule yields the following ranges:
- 68% of data falls between \( 92 - 7 \) and \( 92 + 7 \): \( 85 < X < 99 \)
- 95% of data falls between \( 92 - 14 \) and \( 92 + 14 \): \( 78 < X < 106 \)
- 99.7% of data falls between \( 92 - 21 \) and \( 92 + 21 \): \( 71 < X < 113 \)

Central Limit Theorem (CLT)

The Central Limit Theorem posits that the distribution of the sample mean will approximate a normal distribution, regardless of the original population distribution, as the sample size \( n \) becomes large (usually \( n > 30 \) is considered adequate).
For a sample size of \( n = 10 \) with a standard deviation of \( \sigma = 8.97\), the standard error of the mean (SEM) is calculated as:
\[ SEM = \frac{\sigma}{\sqrt{n}} = \frac{8.97}{\sqrt{10}} \approx 2.84 \]
For proportions, given a success proportion of \( p = 0.5 \) and \( n = 30 \):
\[ SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.5(0.5)}{30}} \approx 0.0913 \]

Descriptive Statistics

Descriptive statistics can summarize and provide insight into a data set. For example, using the following values:
Mean = 4, Median = 4, Mode = 1, Population variance = 11, and Standard deviation = 3.
The sample size \( n \) is 9, with a range calculated as:
\[ \text{Range} = \text{Max} - \text{Min} = 11 - 1 = 10 \]
Interquartile Range (IQR) indicates the spread of the middle 50% of the data, namely:
\[ IQR = Q_3 - Q_1 = 7.5 - 1.5 = 6 \]

Conclusion

Understanding and applying normal probability, the Empirical Rule, the Central Limit Theorem, and descriptive statistics are integral to the field of statistics. These fundamental concepts help researchers, data analysts, and organizations comprehend data distributions and make informed decisions.

References

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