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Statistics : Confidence Interval Tutorials

Confidence Interval for Proportions

  • STATISTICS
CI for difference between two population proportions Let $X_1$ be the observed number of individuals possessing certain attributes (number of successes) in a random sample of size $n_1$ from a large population with population proportion $p_1$ and let $X_2$ be the observed number of individuals possessing certain attributes (number of successes) in a random sample of size $n_2$ from a large population with population proportion $p_2$. The two sample are independent.
2 min Read 1

Confidence Interval for Proportions examples

  • STATISTICS
CI for difference between two population proportions In this tutorial we will discuss some examples on confidence interval for difference between two population proportions. Example 1 A random sample of 85 parts manufactured by machine A yields 10 defective and a random sample of 110 parts manufactured by machine B shows 28 defective. Compute 95% confidence interval for difference between the proportions of defectives. Are the two machine differ significantly with respect to the proportion of defectives?
6 min Read 4

Confidence Interval for Ratio of variances

  • STATISTICS
Confidence Interval for ratio of variances Let $X_1, X_2, \cdots , X_{n_1}$ be a random sample of size $n_1$ from $N(\mu_1, \sigma_1^2)$ and $Y_1, Y_2, \cdots , Y_{n_2}$ be a random sample of size $n_2$ from $N(\mu_2, \sigma_2^2)$. Moreover, $X$ and $Y$ are independently distributed. $100(1-\alpha)$% confidence interval estimate for the ratio of variances is $$ \begin{aligned} \bigg(\frac{s_1^2}{s_2^2}\frac{1}{F_{(\alpha/2,n_1-1,n_2-1)}}, \frac{s_1^2}{s_2^2}\frac{1}{F_{(1-\alpha/2,n_1-1,n_2-1)}}\bigg) \end{aligned} $$ Assumptions a. The two populations are independent. b. The two samples are simple random samples.
1 min Read 0

Confidence Interval for Ratio of variances examples

  • STATISTICS
Confidence Interval for ratio of variances In this tutorial we will discuss some examples on confidence interval for ratio of variances. Example 1 The same capacity of hard drive is manufactured on two different machines, Machine A and Machine B. Samples are taken from both machines and sample mean manufacturing times and sample variances are recorded as follows: Machine A: $n_1=13$, $\overline{x}= 127.4$, $s_1^2= 384.16$ Machine B: $n_2=9$, $\overline{y}= 108.3$, $s_2^2 =106.
3 min Read 2

Confidence Interval for Variance

  • STATISTICS
Introduction Confidence interval can be used to estimate the population parameter with the help of an interval with some degree of confidence. One such a parameter that can be estimated is a population variance or population standard deviation. In this article we will discuss step by step procedure to construct a confidence interval for population variance or population standard deviation. Confidence Interval for Variance Let $X_1, X_2, \cdots , X_n$ be a random sample of size $n$ from $N(\mu, \sigma^2)$.
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Confidence Interval for Variance examples

  • STATISTICS
Confidence Interval for Variance In this tutorial we will discuss some numerical examples to understand how to construct a confidence interval for population variance or population standard deviation. Example 1 The mean replacement time for a random sample of 12 microwaves is 8.6 years with a standard deviation of 3.6 years. Construct a 95% confidence interval for the population standard deviation. Solution Here the sample size is $n=27$, sample standard deviation is $s=6.
3 min Read 2

Plus Four Confidence Interval for Proportion

  • STATISTICS
Introduction In this article we will discuss step by step procedure to construct a plus four confidence interval for population proportion. The plus four confidence interval for the population proportion can be used when the confidence coefficient is more than 90% and the sample size of the population is at least 10. Plus Four Confidence Interval for Proportion Let $X$ be the observed number of individuals possessing certain attributes (number of successes) in a random sample of size $n$ from a large population with population proportion $p$.
2 min Read 1

Plus Four Confidence Interval for Proportion Examples

  • STATISTICS
Plus Four Confidence Interval for Proportion In this article we will discuss step by step examples to construct a plus four confidence interval for population proportion. Example 1 Using Plus Four Confidence Interval for Proportion In a random sample of 60 students from a college, 32 opted mathematics as major subject. Using plus four method find a 95% confidence interval for the proportion of students who opted mathematics as a major subject.
4 min Read 3

Plus Four Confidence Interval for Proportions

  • STATISTICS
Plus Four CI for difference between two population proportions The plus four method produces more accurate confidence interval for difference between two population proportions. We assume that we had four additional observations. One success and one failure in each of the two samples. The plus four method when the confidence coefficient is at least 90% and both the sample sizes are at least 5. Let $X_1$ be the observed number of individuals possessing certain attributes (number of successes) in a random sample of size $n_1$ from a large population with population proportion $p_1$ and let $X_2$ be the observed number of individuals possessing certain attributes (number of successes) in a random sample of size $n_2$ from a large population with population proportion $p_2$.
3 min Read 1

Plus Four Confidence Interval for Proportions Examples

  • STATISTICS
Plus Four CI for difference between two population proportions In this tutorial we will discuss some step by step numerical examples to estimate plus four confidence interval for the difference between two population proportions. Example For two independent samples A and B following information is available: Sample A has 790 number of successes from a sample size of 805. Sample B has 798 number of successes from a sample size of 811.
2 min Read 1

Confidence Interval for means t unequal variances

  • STATISTICS
Confidence interval for means (variances are unknown and unequal) Let $x_1, x_2, \cdots, x_{n_1}$ be a random sample of size $n_1$ from a population with mean $\mu_1$ and standard deviation $\sigma_1$. Let $y_1, y_2, \cdots, y_{n_2}$ be a random sample of size $n_2$ from a population with mean $\mu_2$ and standard deviation $\sigma_2$. And the two sample are independent. Let $\overline{x} = \frac{1}{n_1}\sum x_i$ and $\overline{y} = \frac{1}{n_2}\sum y_i$ be the sample means of first and second sample respectively.
3 min Read 1

Confidence interval for means t unequal variances examples

  • STATISTICS
CI for difference between two population means with unknown and unequal variances In this tutorial we will discuss step by step solution to some numerical problems on confidence interval for difference in means when the population variances are unknown and unequal. Example 1 To compare the mean lifespans of African elephants in the wild and in a zoo, you randomly select several lifespans from both locations. The results are given below:
4 min Read 3