# 7.4 - Confidence Intervals for Means

Previously we considered confidence intervals for 1-proportion and our multiplier in our interval used a *z*-value. But what if our variable of interest is a quantitative variable (e.g. GPA, Age, Height) and we want to estimate the population mean? In such a situation proportion confidence intervals are not appropriate since our interest is in a **mean** amount and not a proportion.

We apply similar techniques but now we are interested in estimating the population mean (\(\mu\)) by using the sample statistic (\(\overline{x}\)) and the multiplier is a *t* value. At the end of Lesson 6 you were introduced to this *t* distribution. Similar to the *z* values that you used as the multiplier for constructing confidence intervals for population proportions, here you will use *t* values as the multipliers. Because *t* values vary depending on the number of degrees of freedom (df), you will need to use either the *t* table or statistical software to look up the appropriate *t* value for each confidence interval that you construct. Using either method, the degrees of freedom will be based on the sample size, *n*. Since we are working with one sample here, \(df=n-1\).

### Finding the Multiplier

Reading the *t *table is slightly more complicated than reading the *z *table because for each different degree of freedom there is a different distribution. In order to locate the correct multipler on the *t *table you will need two pieces of information: (1) the degrees of freedom and (2) the confidence level. The columns of the *t* table are for different confidence levels (80%, 90%, 95%, 98%, 99%, 99.8%). The rows of the *t* table are for different degrees of freedom. The multiplier is at the intersection of the two.

### Examples

**Cups of Coffee**

A research team wants to estimate the number of cups of coffee the average Penn State student consumes each week with 95% confidence. They take a random sample of 20 students and ask how many cups of coffee they drink each week.

Our confidence level is 95%. \(df=n-1=20-1=19\)

Using the *t* table, our multiplier will be 2.093

**Average Height**

Baseball analysts are studying the average height of pitchers. They take a random sample of 55 pitchers and measure their height. They want to construct a 98% confidence interval.

Our confidence level is 98%. \(df=55-1=54\)

Our *t* table does not provide us with multipliers for 54 degrees of freedom. To be more conservative, we will use 50 degrees of freedom because that will give us the larger multiplier.

Using the *t* table, our multiplier will be 2.403

You can also use statistical software to look up *t* multipliers.

### Finding *t*-Multipliers with Minitab Express and Minitab

To find the *t*-multipliers in Minitab Express:

- Probability > Probability Distribution > Display Probability
- Select
*t*distribution and enter your degrees of freedom - Select "A Specified Probability" and "Equal Tails"
- The probability is equal to alpha (i.e., \(1 - confidence \;level\))
- Click Ok, the values at the bottom of the graph (seen below) are your multipliers.

To find the *t*-multipliers in Minitab:

- Graph > Probability Distirbution Plot > View Probability
- Change "Distribution" to
*t*and enter your degrees of freedom - Click the "Shaded Area" tab and select "Both Tails," the proportion in both tails will be equal to alpha (i.e., \(1 - confidence \;level\))
- For example, for a 95% confidence interval, the proportion in both tails would equal \(1-.95=.05\). You would enter .05
- Click Ok, the values at the bottom of the graph are your multipliers.

### Constructing a Confidence Interval for \(\mu\)

Let’s review some of symbols and equations that we learned in previous lessons:

Sample size | \(n\) |

Population mean | \(\mu=\frac{\sum X}{N}\) |

Sample mean | \(\overline{x}= \frac{\sum x}{n}\) |

Standard error of \(\mu\) | \(SE(\overline{x})=\frac{s}{\sqrt{n}}\) |

Multiplier | \(t^{*} \) |

Degrees of freedom (one group) | \(df=n-1\) |

Recall from earlier this lesson, the general form for a confidence interval is \(point\;estimate\pm (standard\;error)(multiplier)\)

For a population mean, the point estimate is \(\overline{x}\), the standard error is \(SE(\overline{x})\) and the multiplier is \(t^{*} \). When we put these together, the formula for a confidence interval for a population mean is

**Confidence Interval for a Population Mean**

\(\overline{x} \pm t^{*} \frac{s}{\sqrt{n}}\)

For large sample sizes, (\(n\geq 30\)), using \(t^{*}=2\) gives an approximate 95% confidence interval.

### Example:** Sleep Deprivation**

In a class survey, students are asked how many hours they sleep per night. In the sample of 22 students, the mean was 5.77 hours with a standard deviation of 1.572 hours. Let’s construct a 95% confidence interval for the mean number of hours slept per night in the population from which this sample was drawn.

This is what we know: \(n=22\), \(\overline{x}=5.77\), and \(s=1.572\).

In order to compute the confidence interval for \(\mu\) we will need the *t* multiplier and the standard error (\( \frac{s}{\sqrt{n}}\)).

\(df=n-1=22-1=21\)

For a 95% confidence interval with 21 degrees of freedom, \(t^{*}=2.080\)

\(SE(\overline{x})=\frac{s}{\sqrt{n}}=\frac{1.572}{\sqrt{22}}=0.335\)

Thus, our confidence interval for \(\mu\) is: \(5.77\pm 2.080(0.335)=5.77\pm0.697=[5.073,\;6.467]\)

We are 95% confident that the population mean is between 5.073 and 6.467 hours.

**What if we wanted to be more conservative and construct a 99% confidence interval?**

The only thing that would change is our multiplier. Now, \(t^{*}=2.831\).

\(5.77\pm 2.831(0.335)=5.77\pm0.948=[4.822,\;6.718]\)

We are 99% confident that the population mean is between 4.822 and 6.718 hours.

### Example:** Milk Production**

A study of 66,831 dairy cows found that the mean milk yield was 12.5 kg per milking with a standard deviation of 4.3 kg per milking (data from Berry, et al., 2013). Construct a 95% confidence interval for the average milk yield in the population.

\(SE(\overline{x})=\frac{s}{\sqrt{n}}=\frac{4.3}{\sqrt{66831}}=0.0166\)

The standard error is small because the sample size is very large.

\(df=66831-1=66830\)

As degrees of freedom approach infinity, the *t* distribution approaches the *z* distribution. Our *t* table only goes to \(df=100\), so we can use the last line where \(df=infinity\).

\(t^{*}=1.96\)

95% C.I.: \(12.5\pm1.96(0.017)=12.5\pm0.033=[12.467,\;12.533]\)

We are 95% confident that the mean milk yield in the population is between 12.467 and 12.533 kg per milking.

### Finding the Sample Size for Estimating a Population Mean

Calculating the sample size necessary for estimating a population mean with a given margin of error and level of confidence is similar to that for estimating a population proportion. However, since the *t* distribution is not as “neat” as the standard normal distribution, the process can be iterative. (Recall, the shape of the *t* distribution is different for each degree of freedom). This means that we would solve, reset, solve, reset, etc. until we reached a conclusion. Yet, we can avoid this iterative process if we employ an approximate method based on *t* distribution approaching the standard normal distribution as the sample size increases. This approximate method invokes the following formula:

**Finding the Sample Size for Estimating a Population Mean**

\[n=\frac{z^{2}s^{2}}{M^{2}}\]

\(z\) = z value for given confidence level\(s\) = sample standard deviation

\(M\) = margin of error

The sample standard deviation may be estimated on the basis of prior research studies.

### Example: Estimating IQ

A team of researchers wants to estimate the mean IQ of students enrolled at one prestigious university. Previous research studies have examined samples of students from other similar universities and usually find results around \(\overline{x}=120\) and \(s=10\). In order to construct a 90% confidence interval with a margin of error of \(\pm2\) IQ points, what sample size should be obtained?

The z value associated with a 90% confidence interval is 1.645. The estimated sample standard deviation is 10. The desired margin of error is 2.

\(n=\frac{z^{2}s^{2}}{M^{2}}=\frac{1.645^{2}(10^{2})}{2^{2}}=67.615\)

We round up to 68. The research team should attempt to obtain a sample of at least 68 individuals.

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Berry, D. P., Coyne, J., Boughlan, B., Burke, M., McCarthy, J., Enright, B., Cromie, A. R., McParland, S. (2013). Genetics of milking characteristics in dairy cows. Animal, 7(11), 1750-1758.