Continuity Correction- adjustment made when a discrete distribution is approximated by a continuous distribution.

Practice Problems

\(\textbf{1)}\) Using a normal approximation to the binomial,
find \(P(X=20)\) for \(n=30\) and \(p=0.6\).

Show Answer

The answer is \(P(X=20)\approx 0.1123\)

Show Work

\(\text{Check if normal approximation is appropriate}\)
\(\,\,\,\,\,\,np\ge 5,\,\,\,nq\ge 5\)
\(\,\,\,\,\,\,(30)(0.6)=18\ge 5,\,\,\,(30)(0.4)=12\ge 5\)
\(\text{Find the mean and standard deviation}\)
\(\,\,\,\,\,\,\mu=(n)(p)=(30)(0.6)=18\)
\(\,\,\,\,\,\,\sigma=\sqrt{npq}=\sqrt{(30)(0.6)(0.4)} \approx 2.683\)
\(\text{Question}\)
\(\,\,\,\,\,\,P(X=20)\)
\(\text{Continuity Correction}\)
\(\,\,\,\,\,\,P(19.5 \le X \le 20.5)\)
\(\text{Find z-scores}\)
\(z=\frac{X-\mu}{\sigma}\)
\(z_1 \approx \frac{19.5-18}{2.683} \approx 0.559\)
\(z_2 \approx \frac{20.5-18}{2.683} \approx 0.932\)
\(\,\,\,\,\,\,P(0.559 \le z \le 0.932)\)
\(\text{Find z values}\)
\(\,\,\,\,\,\,P(z \le 0.559) \approx 0.7119\)
\(\,\,\,\,\,\,P(z \le 0.932) \approx 0.8243\)
\(\,\,\,\,\,\,P(0.559 \le z \le 0.932) \approx 0.8243-0.7119 \approx 0.1124\)
\(\text{Conclusion}\)
\(\,\,\,\,\,\,P(X=20) \approx 0.1124\)

\(\textbf{2)}\) Using a normal approximation to the binomial,
find \(P(X=10)\) for \(n=20\) and \(p=0.7\).

Show Answer

The answer is \(P(X=10)\approx 0.0298\)

Show Work

\(\text{Check if normal approximation is appropriate}\)
\(\,\,\,\,\,\,np\ge 5,\,\,\,nq\ge 5\)
\(\,\,\,\,\,\,(20)(0.7)=14\ge 5,\,\,\,(20)(0.3)=6\ge 5\)
\(\text{Find the mean and standard deviation}\)
\(\,\,\,\,\,\,\mu=(n)(p)=(20)(0.7)=14\)
\(\,\,\,\,\,\,\sigma=\sqrt{npq}=\sqrt{(20)(0.7)(0.3)} \approx 2.0494\)
\(\text{Question}\)
\(\,\,\,\,\,\,P(X=10)\)
\(\text{Continuity Correction}\)
\(\,\,\,\,\,\,P(9.5 \le X \le 10.5)\)
\(\text{Find z-scores}\)
\(z=\frac{X-\mu}{\sigma}\)
\(z_1 \approx \frac{9.5-14}{2.0494} \approx -2.196\)
\(z_2 \approx \frac{10.5-14}{2.0494} \approx -1.708\)
\(\,\,\,\,\,\,P(-2.196 \le z \le -1.708)\)
\(\text{Find z values}\)
\(\,\,\,\,\,\,P(z \le -2.196) \approx 0.0141\)
\(\,\,\,\,\,\,P(z \le -1.708) \approx 0.0438/latex]
[latex]\,\,\,\,\,\,P(-2.196 \le z \le -1.708) \approx 0.0438-0.0141 \approx 0.0298\)
\(\text{Conclusion}\)
\(\,\,\,\,\,\,P(X=10) \approx 0.0298\)

 

See Related Pages\(\)

\(\bullet\text{ Statistics Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Uniform Distribution}\)
\(\,\,\,\,\,\,\,\,p(x)=\frac{1}{b-a}…\)

\(\bullet\text{ Binomial Distribution}\)
\(\,\,\,\,\,\,\,\,p(r)={}_{n}C_{r}(p)^r(1-p)^{n-r}…\)

\(\bullet\text{ Poisson Distribution}\)
\(\,\,\,\,\,\,\,\,P(x)=\displaystyle\frac{\lambda^x e^{-\lambda}}{x!}…\)

\(\bullet\text{ Geometric Distribution}\)
\(\,\,\,\,\,\,\,\,P(X=n)=p(1-p)^{n-1}…\)

 

Related Pages

\(\bullet\text{ Normal Approximation Calculator (Omnicalculator.com)}\)

 

In Summary

Continuity correction is typically covered in probability and statistics courses. It is a mathematical technique that helps to approximate discrete probability distributions with continuous ones. It is often used when working with small sample sizes or when the probability of an event is close to zero or one, it involves making small adjustments to the probabilities of events in order to account for the fact that they are not perfectly continuous. These adjustments are made in order to more accurately reflect the true probability of an event occurring. This technique is useful for a variety of applications in fields such as statistics, finance, and engineering. It can help us to more accurately model and predict the outcome of events, and to make more informed decisions based on these predictions.

Math topics that use continuity correction

Here are five other math topics that use continuity correction in probability:

Central limit theorem: The central limit theorem states that the distribution of the sum of a large number of independent and identically distributed (iid) random variables approaches a normal distribution as the number of variables increases. Continuity correction is often used to approximate the distribution of the sum when the number of variables is not very large.

Normal distribution: The normal distribution is a continuous probability distribution that is symmetric around the mean. It is often used to model continuous data and is characterized by its mean and standard deviation. Continuity correction can be used to approximate probabilities involving the normal distribution when working with discrete data.

Binomial distribution: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials (trials with only two possible outcomes). Continuity correction can be used to approximate probabilities involving the binomial distribution when working with continuous data.

Poisson distribution: The Poisson distribution is a discrete probability distribution that models the number of times an event occurs in a fixed time period or space. It is often used to model rare events, such as the number of customers arriving at a store in a given hour. Continuity correction can be used to approximate probabilities involving the Poisson distribution when working with continuous data.

Hypothesis testing: Continuity correction is often used in statistical hypothesis testing, particularly in testing for the difference between two means. It can be used to adjust the test statistic in order to account for the fact that the distribution of the test statistic is continuous, even though the underlying data may be discrete.