Using Normal Distributions to Estimate Populations - Statistics and Probability

1. Using the Empirical Rule, estimate $P(X<\mu-3\sigma)$ for normal data.

   About $0.0015$. 99.7% within $3\sigma$, so 0.3% in both tails, 0.15% in lower tail.

2. What is the mean and standard deviation of the standard normal distribution?

   Mean $0$, standard deviation $1$. Standard normal is centered at 0 with unit spread.

3. What does the notation $\Phi(z)$ represent in normal distribution tables?

   $\Phi(z)=P(Z\le z)$ for $Z\sim N(0,1)$. Cumulative distribution function for standard normal.

4. Find $x$ for $z=-2$ in a normal model with $\mu=50$ and $\sigma=4$.

   $x=42$. Apply $x=50+(-2)(4)=50-8$.

5. What percentage of data is within $3\sigma$ of the mean in an approximately normal distribution?

   About $99.7\%$. Third part of the 68-95-99.7 rule for normal distributions.

6. Which condition is required to estimate percentages using a normal model: strong skew, outliers, or roughly bell-shaped?

   Roughly bell-shaped and symmetric with no strong outliers. Normal model requires symmetric, unimodal data.

7. What is the probability expression for being between two $z$-scores $a$ and $b$ in $N(0,1)$?

   $P(a<Z<b)=\Phi(b)-\Phi(a)$. Interval probability equals difference of cumulative probabilities.

8. Which situation makes a normal model inappropriate: unimodal symmetric data or strongly right-skewed data?

   Strongly right-skewed data. Skewed distributions violate normality assumption.

9. What is the 68-95-99.7 rule percentage within $1\sigma$ of the mean for a normal model?

   About $68\%$. First part of the empirical rule for normal distributions.

10. What proportion of a normal distribution lies above the mean $\mu$?

    $0.50$. Normal distribution is symmetric about the mean.

11. Using the Empirical Rule, estimate $P(X>\mu+2\sigma)$ for normal data.

    About $0.025$. 95% within $2\sigma$, so 5% in both tails, 2.5% in upper tail.

12. Which plot is most commonly used to check whether a data set is approximately normal: dotplot, histogram, or boxplot?

    Histogram. Shows distribution shape to assess normality visually.

13. What percentage of data is within $1\sigma$ of the mean in an approximately normal distribution?

    About $68\%$. First part of the 68-95-99.7 rule for normal distributions.

14. What is the empirical rule estimate for the proportion between $\mu$ and $\mu+1\sigma$?

    About $34\%$. Half of $68\%$ lies between mean and one SD above.

15. What is the $z$-score formula for a value $x$ given mean $\mu$ and standard deviation $\sigma$?

    $z=\frac{x-\mu}{\sigma}$. Measures how many standard deviations a value is from the mean.

16. Using a $z$-table where $\Phi(0.80)=0.7881$, what is $P(-0.80<Z<0.80)$?

    $0.5762$. By symmetry: $2(0.7881)-1$ or $0.7881-(1-0.7881)$.

17. What is the $z$-score formula for a data value $x$ with mean $\mu$ and standard deviation $\sigma$?

    $z=\frac{x-\mu}{\sigma}$. Standardizes values by subtracting mean and dividing by standard deviation.

18. Find $z$ for $x=85$, $\mu=70$, $\sigma=5$.

    $z=3$. $z=\frac{85-70}{5}=\frac{15}{5}=3$

19. What is the 68-95-99.7 rule percentage within $2\sigma$ of the mean for a normal model?

    About $95\%$. Second part of the empirical rule for normal distributions.

20. Using the empirical rule, estimate $P(\mu-1\sigma<X<\mu+1\sigma)$ for a normal model.

    About $68\%$. Direct application of the $68\%$ empirical rule.

21. What is the probability formula for being between two $z$-scores $a$ and $b$ using the normal CDF $\Phi$?

    $P(a<Z<b)=\Phi(b)-\Phi(a)$. Difference of CDFs gives probability between two values.

22. If a table gives $\Phi(1.25)=0.8944$, what is $P(Z>1.25)$?

    $0.1056$. $P(Z>1.25)=1-0.8944=0.1056$

23. What is the formula to convert a $z$-score to a data value $x$ using $\mu$ and $\sigma$?

    $x=\mu+z\sigma$. Reverses the z-score formula to find the original data value.

24. What is the complement rule for a standard normal probability $P(Z>z)$ in terms of $P(Z\le z)$?

    $P(Z>z)=1-P(Z\le z)$. Uses the complement rule for probability.

25. Identify the correct condition: When is fitting a normal model to data generally inappropriate?

    When data are strongly skewed or have outliers. Normal models assume symmetric, bell-shaped data.

26. Identify the distribution model used to fit data with mean $\mu$ and standard deviation $\sigma$ as normal.

    $X\sim N(\mu,\sigma)$. Standard notation for normal distribution with parameters.

27. Using the Empirical Rule, estimate $P(\mu-\sigma<X<\mu+\sigma)$ for normal data.

    About $0.68$. 68% rule: about 68% within one standard deviation.

28. Identify the correct condition: What plot shape most supports using a normal model for a data set?

    A roughly symmetric, unimodal, bell-shaped distribution. These characteristics match the normal distribution shape.

29. Find $z$ for $x=85$ in a normal model with $\mu=70$ and $\sigma=10$.

    $z=1.5$. Apply $z=\frac{85-70}{10}=\frac{15}{10}$.

30. What is the inverse conversion from a $z$-score to a data value $x$ for mean $\mu$ and SD $\sigma$?

    $x=\mu+z\sigma$. Reverses the z-score formula to find the original data value.

31. Find $x$ when $z=-1.5$, $\mu=100$, and $\sigma=20$.

    $x=70$. $x=100+(-1.5)(20)=100-30=70$

32. What symmetry rule relates $P(Z\le -z)$ and $P(Z\le z)$ for $Z\sim N(0,1)$?

    $P(Z\le -z)=1-\Phi(z)$. Uses symmetry of normal curve around mean 0.

33. Using the empirical rule, estimate $P(X>\mu+2\sigma)$ for a normal model.

    About $2.5\%$. $(100\%-95\%)/2$ gives the upper tail beyond $2\sigma$.

34. What is the symmetry rule for normal models relating $P(Z\le z)$ and $P(Z\ge -z)$?

    $P(Z\le z)=P(Z\ge -z)$. Normal distribution is symmetric about zero for standard normal.

35. What is the 68-95-99.7 rule percentage within $3\sigma$ of the mean for a normal model?

    About $99.7\%$. Third part of the empirical rule for normal distributions.

36. What is the complement rule for a standard normal probability such as $P(Z>z)$?

    $P(Z>z)=1-\Phi(z)$. Right tail probability equals 1 minus left tail probability.

37. Using a $z$-table where $\Phi(1.25)=0.8944$, what is $P(Z>1.25)$?

    $0.1056$. Apply complement rule: $1-0.8944$.

38. What percentage of data is within $2\sigma$ of the mean in an approximately normal distribution?

    About $95\%$. Second part of the 68-95-99.7 rule for normal distributions.

39. What is the empirical rule estimate for the proportion between $\mu+1\sigma$ and $\mu+2\sigma$?

    About $13.5\%$. $(95\%-68\%)/2$ gives this tail proportion.

40. What is the empirical rule estimate for the proportion between $\mu+2\sigma$ and $\mu+3\sigma$?

    About $2.35\%$. $(99.7\%-95\%)/2$ gives this tail proportion.

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