A 99 percent confidence interval would be wider than a 95 percent confidence interval (for example, plus or minus 4.5 percent instead of 3.5 percent). A 90 percent confidence interval would be narrower (plus or minus 2.5 percent, for example).
Minitab calculates that the 95% confidence interval is 1230 – 1265 hours. The confidence interval indicates that you can be 95% confident that the mean for the entire population of light bulbs falls within this range. As you increase the sample size, the sampling error decreases and the intervals become narrower.
For example, a 99% confidence interval will be wider than a 95% confidence interval because to be more confident that the true population value falls within the interval we will need to allow more potential values within the interval. The confidence level most commonly adopted is 95%.
How to Construct a Confidence Interval
- Identify a sample statistic. Choose the statistic (e.g, sample mean, sample proportion) that you will use to estimate a population parameter.
- Select a confidence level.
- Find the margin of error.
- Specify the confidence interval.
The t test tells you how significant the differences between groups are; In other words it lets you know if those differences (measured in means/averages) could have happened by chance. Another example: Student's T-tests can be used in real life to compare means.
To compute the 95% confidence interval, start by computing the mean and standard error: M = (2 + 3 + 5 + 6 + 9)/5 = 5. σM = = 1.118. Z.95 can be found using the normal distribution calculator and specifying that the shaded area is 0.95 and indicating that you want the area to be between the cutoff points.
So, if your significance level is 0.05, the corresponding confidence level is 95%. If the P value is less than your significance (alpha) level, the hypothesis test is statistically significant. If the confidence interval does not contain the null hypothesis value, the results are statistically significant.
Providing a Range of Values
You determine the level of confidence, but it is generally set at 90%, 95%, or 99%. Confidence intervals use the variability of your data to assess the precision or accuracy of your estimated statistics.For an approximately normal data set, the values within one standard deviation of the mean account for about 68% of the set; while within two standard deviations account for about 95%; and within three standard deviations account for about 99.7%.
Generally, the rule of thumb is that the larger the sample size, the more statistically significant it is—meaning there's less of a chance that your results happened by coincidence.
Importance of Confidence Intervals. Market research is about reducing risk. Confidence intervals are about risk. They consider the sample size and the potential variation in the population and give us an estimate of the range in which the real answer lies.
Calculating the Confidence Interval
| Confidence Interval | Z |
|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
| 99.5% | 2.807 |
How to Find a Sample Size Given a Confidence Interval and Width (unknown population standard deviation)
- za/2: Divide the confidence interval by two, and look that area up in the z-table: .95 / 2 = 0.475.
- E (margin of error): Divide the given width by 2. 6% / 2.
- : use the given percentage. 41% = 0.41.
- : subtract. from 1.
An odds ratio of exactly 1 means that exposure to property A does not affect the odds of property B. An odds ratio of more than 1 means that there is a higher odds of property B happening with exposure to property A. An odds ratio is less than 1 is associated with lower odds.
Statistics For Dummies, 2nd Edition
| Confidence Level | z*– value |
|---|
| 80% | 1.28 |
| 85% | 1.44 |
| 90% | 1.64 |
| 95% | 1.96 |
Since the z-score is the number of standard deviations above the mean, z = (x - mu)/sigma. Solving for the data value, x, gives the formula x = z*sigma + mu. So the data value equals the z-score times the standard deviation, plus the mean.
To find a critical value, look up your confidence level in the bottom row of the table; this tells you which column of the t-table you need. Intersect this column with the row for your df (degrees of freedom). The number you see is the critical value (or the t*-value) for your confidence interval.
The lower confidence limit is 45.3 (70.0−24.7), and the upper confidence limit is 94.7 (70+24.7).
3) a) A 90% Confidence Interval would be narrower than a 95% Confidence Interval. This occurs because the as the precision of the confidence interval increases (ie CI width decreasing), the reliability of an interval containing the actual mean decreases (less of a range to possibly cover the mean).
Increasing the sample size decreases the width of confidence intervals, because it decreases the standard error. c) The statement, "the 95% confidence interval for the population mean is (350, 400)", is equivalent to the statement, "there is a 95% probability that the population mean is between 350 and 400".
A 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population. With large samples, you know that mean with much more precision than you do with a small sample, so the confidence interval is quite narrow when computed from a large sample.
