7.7: t-Interval for a Mean (2024)

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    7.7.1 Student’s T-Distribution

    A t-distribution is another symmetric distribution for a continuous random variable.

    7.7: t-Interval for a Mean (2)

    Gosset

    William Gosset was a statistician employed at Guinness and performed statistics to find the best yield of barley for their beer. Guinness prohibited its employees to publish papers so Gosset published under the name Student. Gosset’s distribution is called the Student’s t-distribution.

    A t-distribution is another special type of distribution for a continuous random variable.

    Properties of the t-distribution density curve:

    1. Symmetric, Unimodal (one mode) Bell-shaped.
    2. Centered at the mean μ = median = mode = 0.
    3. The spread of a t-distribution is determined by the degrees of freedom which are determined by the sample size.
    4. As the degrees of freedom increase, the t-distribution approaches the standard normal curve.
    5. The total area under the curve is equal to 1 or 100%.

    7.7: t-Interval for a Mean (3)

    Figure 7-7

    Figure 7-7 shows examples of three different t-distributions with degrees of freedom of 1, 5 and 30. Note that as the degrees of freedom increase the distribution has a smaller standard deviation and will get closer in shape to the normal distribution.

    The t-critical value that has 5% of the area in the upper tail for n = 13.

    Solution

    Use a t-distribution with the degrees of freedom, df = n – 1 = 13 – 1 = 12. Draw and shade the upper tail area as in Figure 7-8. Use the DISTR menu invT option. Note that if you have an older TI-84 or a TI-83 calculator you need to have the program INVT installed.

    For this function, you always use the area to the left of the point. If want 5% in the upper tail, then that means there is 95% in the bottom tail area. tα = invT(area below t-score, df) = invT(0.95,12) = 1.782

    7.7: t-Interval for a Mean (4)

    7.7: t-Interval for a Mean (5)

    You can download the INVT program to your calculator from http://MostlyHarmlessStatistics.com or use Excel =T.INV(0.95,12) = 1.7823.

    Compute the probability of getting a t-score larger than 1.8399 with a sample size of 13.

    Solution

    To find the P(t > 1.8399) on the TI calculator, go to DISTR use tcdf(lower,upper,df). For this example, we would have tcdf(1.8399,∞,12). In Excel use =1-T.DIST(1.8399,12,TRUE) = 0.0453. P(t > 1.8399) = 0.0453.

    7.7: t-Interval for a Mean (6)

    Figure 7-9

    7.7: t-Interval for a Mean (7)

    7.7.2 T-Confidence Interval

    Note that we rarely have a calculation for the population standard deviation so in most cases we would need to use the sample standard deviation as an estimate for the population standard deviation. If we have a normally distributed population with an unknown population standard deviation then the sampling distribution of the sample mean will follow a t-distribution.

    7.7: t-Interval for a Mean (8)

    Figure 7-10

    A 100(1 - \(\alpha\))% Confidence Interval for a Population Mean μ: (σ unknown)

    Choose a simple random sample of size n from a population having unknown mean μ.

    The 100(1 - \(\alpha\))% confidence interval estimate for μ is given by \(\bar{x} \pm t_{\alpha / 2, n-1}\left(\frac{s}{\sqrt{n}}\right)\).

    The df = degrees of freedom* are n – 1.

    The degrees of freedom are the number of values that are free to vary after a sample statistic has been computed. For example, if you know the mean was 50 for a sample size of 4, you could pick any 3 numbers you like, but the 4th value would have to be fixed to have the mean come out to be 50. For this class we just need to know that degrees of freedom will be based on the sample size.

    The sample mean \(\bar{x}\) is the point estimate for μ, and the margin of error is \(t_{\alpha / 2}\left(\frac{s}{\sqrt{n}}\right)\). Where t\(\alpha\)/2 is the positive critical value on the t-distribution curve with df = n – 1 and area 1 – \(\alpha\) between the critical values –t\(\alpha\)/2 and +t\(\alpha\)/2, as shown in Figure 7-11.

    7.7: t-Interval for a Mean (9)

    Figure 7-11

    Before we compute a t-interval we will practice getting t critical values using Excel and the TI calculator’s built in tdistribution.

    Compute the critical values –t\(\alpha\)/2 and +t\(\alpha\)/2 for a 90% confidence interval with a sample size of 10.

    Solution

    Draw and t-distribution with df = n – 1 = 9, see Figure 7-12. In Excel use =T.INV(lower tail area, df) =T.INV(0.95,9) or in the TI calculator use invT(lower tail area, df) = invT(0.95,9). The critical values are t = ±1.833

    7.7: t-Interval for a Mean (10)

    Figure 7-12

    7.7: t-Interval for a Mean (11)

    We can use Excel to find the margin of error when raw data is given in a problem. The following example is first done longhand and then done using Excel’s Data Analysis Tool and the T-Interval shortcut key on the TI calculator.

    The yearly salary for mathematics assistant professors are normally distributed. A random sample of 8 math assistant professor’s salaries are listed below in thousands of dollars. Estimate the population mean salary with a 99% confidence interval.

    66.0 75.8 70.9 73.9 63.4 68.5 73.3 65.9

    Solution

    First find the t critical value using df = n – 1 = 7 and 99% confidence, t\(\alpha\)/2 = 3.4995.

    7.7: t-Interval for a Mean (12)

    Then use technology to find the sample mean and sample standard deviation and substitute the numbers into the formula.

    \(\bar{x} \pm t_{\alpha / 2, n-1}\left(\frac{s}{\sqrt{n}}\right) \Rightarrow 69.7125 \pm 3.4995\left(\frac{4.4483}{\sqrt{8}}\right) \Rightarrow 69.7125 \pm 5.5037 \Rightarrow(64.2088,75.2162)\)

    The answer can be given as an inequality 64.2088 < µ < 75.2162 or in interval notation (64.2088, 75.2162).

    We are 99% confident that the interval 64.2 and 75.2 contains the true population mean salary for all mathematics assistant professors.

    We are 99% confident that the mean salary for mathematics assistant professors is between $64,208.80 and $75,216.20.

    Assumption: The population we are sampling from must be normal* or approximately normal, and the population standard deviation σ is unknown. *This assumption must be addressed before using statistical inference for sample sizes of under 30.

    TI-84: Press the [STAT] key, arrow over to the [TESTS] menu, arrow down to the [8:TInterval] option and press the [ENTER] key. Arrow over to the [Stats] menu and press the [ENTER] key. Then type in the mean, sample standard deviation, sample size and confidence level, arrow down to [Calculate] and press the [ENTER] key. The calculator returns the answer in interval notation. Be careful. If you accidentally use the [7:ZInterval] option you would get the wrong answer.

    Alternatively (If you have raw data in list one) Arrow over to the [Data] menu and press the [ENTER] key. Then type in the list name, L1, leave Freq:1 alone, enter the confidence level, arrow down to [Calculate] and press the [ENTER] key.

    TI-89: Go to the [Apps] Stat/List Editor, then press [2nd] then F7 [Ints], then select 2:TInterval. Choose the input method, data is when you have entered data into a list previously or stats when you are given the mean and standard deviation already. Type in the mean, standard deviation, sample size (or list name (list1), and Freq: 1) and confidence level, and press the [ENTER] key. The calculator returns the answer in interval notation. Be careful: If you accidentally use the [1:ZInterval] option you would get the wrong answer.

    Excel Directions

    Type the data into Excel. Select the Data Analysis Tool under the Data tab.

    7.7: t-Interval for a Mean (13)

    Select Descriptive Statistics. Select OK.

    7.7: t-Interval for a Mean (14)

    Use your mouse and click into the Input Range box, then select the cells containing the data. If you highlighted the label then check the box next to Labels in first row. In this case no label was typed in so the box is left blank. (Be very careful with this step. If you check the box and do not have a label then the first data point will become the label and all your descriptive statistics will be incorrect.)

    Check the boxes next to Summary statistics and Confidence Level for Mean. Then change the confidence level to fit the question. Select OK.

    The table output does not find the confidence interval. However, the output does give you the sample mean and margin of error.

    The margin of error is the last entry labeled Confidence Level. To find the confidence interval subtract and add the margin of error to the sample mean to get the lower and upper limit of the interval in two separate cells.

    7.7: t-Interval for a Mean (15)

    The following screenshot shows the cell references to find the lower limit as =D3-D16 and the upper limit as =D3+D16. Make sure to put your answer in interval notation.

    7.7: t-Interval for a Mean (16)

    The answer is given as an inequality 64.2088 < µ < 75.2162 or in interval notation (64.2088, 75.2162).

    We are 99% confident that the interval 64.2 and 75.2 contains the true population mean salary for all mathematics assistant professors.

    Summary

    A t-confidence interval is used to estimate an unknown value of the population mean for a single sample. We need to make sure that the population is normally distributed or the sample size is 30 or larger. Once this is verified we use the interval \(\bar{x}-t_{\alpha / 2, n-1}\left(\frac{s}{\sqrt{n}}\right)<\mu<\bar{x}+t_{\alpha / 2, n-1}\left(\frac{s}{\sqrt{n}}\right)\) to estimate the true population mean. Most of the time we will be using the t-interval, not the z-interval, when estimating a mean since we rarely know the population standard deviation. It is important to interpret the confidence interval correctly. A general interpretation where you would change what is in the parentheses to fit the context of the problem is: “One can be 100(1 – \(\alpha\))% confident that between (lower boundary) and (upper boundary) contains the population mean of (random variable in words using context and units from problem).”

    7.7: t-Interval for a Mean (2024)

    FAQs

    7.7: t-Interval for a Mean? ›

    Summary. A t-confidence interval is used to estimate an unknown value of the population mean for a single sample. We need to make sure that the population is normally distributed or the sample size is 30 or larger.

    How to calculate t interval for a mean? ›

    Find the critical value of t in the two-tailed t table. Multiply the critical value of t by s/√n. Add this value to the mean to calculate the upper limit of the confidence interval, and subtract this value from the mean to calculate the lower limit.

    How do you find the mean interval? ›

    To calculate the mean of grouped data, the first step is to determine the midpoint of each interval or class. These midpoints must then be multiplied by the frequencies of the corresponding classes. The sum of the products divided by the total number of values will be the value of the mean.

    What is the t-score for 95 confidence interval? ›

    The t value for 95% confidence with df = 9 is t = 2.262.

    What is the T value of a 95% confidence interval of the population mean based on a sample of 15 observations? ›

    Final answer:

    The t value for a 95% confidence interval of the population mean with a sample of 15 observations is approximately 2.145.

    What is the t-test for a single mean? ›

    The t-test for a single mean enables us to test hypothesis about the population mean when our sample size is small and/or when we do not know the variance of the sampled population.

    What is the t-test for difference in means? ›

    What Is a T-Test? A t-test is an inferential statistic used to determine if there is a significant difference between the means of two groups and how they are related. T-tests are used when the data sets follow a normal distribution and have unknown variances, like the data set recorded from flipping a coin 100 times.

    What is the interval formula? ›

    As we know, class interval = upper limit - lower limit. So, for 0-10, upper limit = 10 and lower limit = 0. Hence, class interval = 10-0 = 10. Similarly, for all classes, 10-20, 20-30, 30-40, 40-50, 50-60.

    How do I find the interval? ›

    How do you find intervals in math? An interval can be found from an inequality. The interval takes the bounds of the inequality as endpoints and uses the inequality symbols to indicate if the endpoints are included in the interval. The values in between the endpoints are defined to be in the interval.

    What is the mean interval estimate? ›

    interval estimation, in statistics, the evaluation of a parameter—for example, the mean (average)—of a population by computing an interval, or range of values, within which the parameter is most likely to be located.

    How to calculate t-value? ›

    The t-score formula is: t = x ― − μ S n , where x ― is the sample mean, μ is the population mean, S is the standard deviation of the sample, and n is the sample size. Remember to square root n in the formula.

    When to use t interval? ›

    T interval is good for situations where the sample size is small and population standard deviation is unknown. When the sample size comes to be very small (n≤30), the Z-interval for calculating confidence interval becomes less reliable estimate.

    How to find the 95% confidence interval? ›

    Since 95% of values fall within two standard deviations of the mean according to the 68-95-99.7 Rule, simply add and subtract two standard deviations from the mean in order to obtain the 95% confidence interval.

    How to calculate 95 confidence interval using t-distribution? ›

    Then a 95% confidence interval (CI) for μ is ˉX±t∗S/√n, where S is the sample standard deviation and where t∗ cuts probability 0.025 from the upper tail of Student's t distribution with n−1 degrees of freedom.

    How do you interpret a 95 confidence interval t test? ›

    A confidence interval indicates where the population parameter is likely to reside. For example, a 95% confidence interval of the mean [9 11] suggests you can be 95% confident that the population mean is between 9 and 11.

    What is the formula for the mean of the t-distribution? ›

    Definition. The Student t -distribution is the distribution of the t -statistic given by t=¯x−μs√n t = x ¯ − μ s n where ¯x is the sample mean, μ is the population mean, s is the sample standard deviation and n is the sample size.

    What is the formula for the t test mean? ›

    t test formula (1 sample) t = M – µ Sx Sample mean (M) minus population mean you are comparing your sample to (µ), divided by the standard error (Sx).

    What is the formula for time interval? ›

    If the times are in the same units (e.g., both in hours), subtract the start time from the end time to find the interval. If the times are in different units (e.g., one in minutes and one in hours), convert them to the same unit before subtracting.

    How do you find the t-score from the mean and standard deviation? ›

    The t-score formula is: t = x ― − μ S n , where x ― is the sample mean, μ is the population mean, S is the standard deviation of the sample, and n is the sample size. Remember to square root n in the formula.

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