The Relationship Between Confidence Intervals & Hypothesis Tests

The Relationship Between Confidence Intervals & Hypothesis Tests
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  • 0:01 Testing a Small Sample
  • 1:20 Setting up the Hypothesis Test
  • 3:44 Developing the…
  • 4:54 Relating Significance…
  • 7:40 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Quantifying population information by testing a small sample is a marvelous mathematical invention. In this lesson, we explore the relationship between confidence intervals and hypothesis tests.

Testing a Small Sample

Grapes have many uses. A partial list includes Cabernet grapes for wine, Thompson grapes for eating raw, Concord grapes for juices, and Muscat grapes for raisins. Growers often use grape weight as a benchmark for quality. Imagine working with Mr. Muscato, the owner of a major Muscat grape distribution company. He wants to track grape weight while his grapes are still growing in the orchards. Obviously, harvesting all the grapes and measuring each one is not practical. Studying a sample of the crop is the way to go.

The grapes in the orchards are the population under study. We want to estimate this grape population's mean weight. We don't know this number, nor do we know its standard deviation. Taking a small number (fewer than 30) of shipping boxes filled with grapes, we can measure the weight of each box of grapes. Muscato tells us that the ideal weight for a box of grapes should be 41.5 pounds.

In this lesson, we won't get into the details of finding a confidence interval or doing hypothesis testing. If you're unfamiliar with either subject, feel free to explore our other lessons that cover each concept in greater detail. Instead, we will explore what the math behind each concept is saying and how the two relate. Also, it's nice at Mr. Muscato's company; there's always a bowl of raisins nearby.

Setting up the Hypothesis Test

Imagine that we take a sample N=12 boxes of grapes and find that the mean weight = 43.5 pounds with a sample standard deviation = 3.947 pounds. Is this close enough to the ideal mean weight of 41.5 pounds?

In hypothesis testing, we set up a null hypothesis. Specifically, we choose as the null hypothesis Ho: the mean of the population μ = 41.5. Basically, this null hypothesis is claiming that there's no significant difference between the population mean weight and the ideal mean weight, which would be good news for Mr. Muscato.

Complementing the null hypothesis is the alternative hypothesis Ha: μ is not equal to 41.5. There are two directions in the alternative hypothesis for which μ may not equal 41.5. It could be less than or greater than 41.5, so we'll perform a two-sided test.

We are hoping to fail to reject the null hypothesis. Otherwise, we might have to alter the growing process. Mr. Muscato reminds us that the population standard deviation is unknown, and the sample size is small. In other words, we can associate the mean with a t-distribution to perform our hypothesis test.

First, we need to choose a significance level, the probability of rejecting the null hypothesis when it's true, usually written as α. We go with α = 0.05. This means there's a risk that 5% of the time we'd wrongly conclude that a difference exists—(reject our null hypothesis) when there is no real difference.

We then find the t-score for α / 2 (since this is a two-sided test) and can use it to find a p-value. This is a probability based on the sample mean. If the p-value is less than or equal to α we decide to reject the null hypothesis. If the p-value is greater than α we fail to reject the null hypothesis.

All we can do with a null hypothesis is reject it or fail to reject it. We can't prove the null hypothesis. While Mr. Muscato hovers nearby, we find that our p-value = 0.1060. This is greater than α of .05, so we fail to reject the null hypothesis. This gives us some encouragement for the quality of the grapes, but we can go further by calculating the confidence interval. Ready for another bowl of raisins?

Developing the Confidence Interval

Using the sample standard deviation, the number of boxes in the sample, the α / 2 t-score, and the sample mean, we can construct a confidence interval. Remember that a confidence interval is a range of values that also expresses uncertainty associated with a population parameter. If μ = 41.5 from the null hypothesis is within the confidence interval, we fail to reject the null hypothesis. Likewise, if the null hypothesis value is outside the confidence interval, we reject the null hypothesis.

Using the values from our hypothesis test, we find the confidence interval CI is [41 46]. Clearly, 41.5 is within this interval so we fail to reject the null hypothesis. This agrees with the results of our hypothesis test. We are 95% confident the population mean weight is between 41 pounds and 46 pounds.

This is good news because the interval contains the ideal mean weight. To celebrate, Mr. Muscato offers free wine to go along with the raisins. Be careful how much you drink though—we still need to relate confidence intervals to hypothesis tests in greater detail!

Relating Significance and P-Value to Confidence Intervals

What happens if we keep everything else the same but rework the confidence interval CI for α levels of .10 and .20? Recall α is a probability. Probabilities have values between 0 and 1. The confidence level is 1 - α. As the probability α increases from 0 to 1, the confidence level decreases from 1 to 0. Take a look at some CIs and p-values with different significance levels α:


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