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4.5 Intervals and Tests for One Mean

Section Overview

  • Interactive 4.5.1 — t-Dist. vs z-Dist.
  • Gallery 4.5.1 — 1 & 2 sided tests for 1 Mean
  • Gallery 4.5.2 — t-table
  • Example 4.5.1
  • Gallery 4.5.3 — Checking Normal Condition
  • Gallery 4.5.4 — 1 Samp. t-test on Calc
  • Interactive 4.5.2 — Example 4.5.1
  • Gallery 4.5.5 — 1 Samp. t-interval on Calc
  • Pop Quiz 4.5.1

A New Type of Distribution

If you’ve been observant, you may have realized at this point how we’ve been using the sample statistic in equations to find the test statistic. So far, we’ve used the sample statistics of the sample proportion p̂ and the difference between two sample proportions p̂₁ − p̂₂ to find the z-statistic. For this section, we know our sample statistic will be the sample mean x̄, but what will the equation and the test statistic be? You may guess our test statistic will be the z-statistic, found by none other than our z-score equation since it contains the sample mean x̄:

z=x−μσz=\frac{x-\mu}{\sigma}
Source gap (S0406): Source formula: the prose refers to the sample mean x̄, while the displayed z-score formula uses x. The original formula is retained.

However, rarely do we ever know the standard deviation, since it is derived from the population. Our solution is to use the t-distribution, a type of distribution created from the t-statistic. The shape of the t-distribution’s density curve is dependent on its degrees of freedom, which is one less than the sample size, or n − 1. The more degrees of freedom the t-distribution has, the more closely it approximates the z-distribution, as shown in Interactive 4.5.1.

Interactive 4.5.1 t-distributions vs z-distribution

Three symmetric density curves centered at μ=0. The high purple curve is labelled z-distribution; the lower lime curve is labelled t-distribution. An annotation Getting closer… points to the intermediate teal curve, showing the t curve approaching the z curve as degrees of freedom increase. Open full-size image S0407

Significance Tests for One Mean

The test we use for the sample mean is the one-sample t test, which calculates the t-statistic:

t=xˉ−μ0sx2/nt=\frac{\bar x-\mu_0}{\sqrt{s_x^2/n}}

where x̄ = sample mean; μ₀ = population mean stated by null hypothesis; sₓ = standard deviation of sample; n = sample size.

Our null hypothesis is: H0:μ=μ0H_0:\mu=\mu_0.

We also have our three possible alternative hypotheses, where:

Ha:μ>μ0orHa:μ<μ0for one-sided testsH_a:\mu>\mu_0\quad\text{or}\quad H_a:\mu<\mu_0\quad\text{for one-sided tests}
Ha:μ≠μ0for two-sided testsH_a:\mu\ne\mu_0\quad\text{for two-sided tests}

See Gallery 4.5.1 for a visualization of one and two-sided tests.

Confidence Intervals for One Mean

The confidence interval we use for the sample mean is the one-sample t-interval, which is calculated as:

Suppose that an SRS of size n₁ is drawn from a Normal population with unknown mean μ₀ and that an The one-sample t-interval is given by

Source gap (S0408): Source wording: the introductory sentence ends “and that an” immediately before “The one-sample t-interval is given by.” No missing continuation is supplied.
xˉ±t∗sx2n\bar x\pm t^*\sqrt{\frac{s_x^2}{n}}

where x̄ = sample mean; sₓ = standard deviation; n = sample size; t* = critical t-value, whose values can be found in the t-table dependent on the distribution’s n − 1 degrees of freedom. The t-table can be found in Gallery 4.5.2.

Normal Condition for Small Sample Sizes

The three conditions needed for performing one-sample t-procedures can be found in Chapter 4.1. However, the Normal Condition warrants further explanation. We already know if the sample size is n ≥ 30, then the one-sample t-procedures can automatically be used due to the Central Limit TheoremEditorial qualification: The captured n ≥ 30 statement is not a universal guarantee. Under the CLT’s assumptions, the accuracy of a Normal approximation depends on the population and sample size.. This holds true even if the distribution is clearly skewed. However, does this spell “game over” if our sample size is less than 30? Not necessarily. For sample sizes n < 15, t-procedures can be used if the data appears close to Normal (no skewness or outliers). For sample sizes 15 ≤ n ≤ 30, the t-procedures can be used as long is there isn’t any strong skewness or outliers.

Source gap (S0412): Source table edges: degrees-of-freedom labels at the left and probability columns beyond p=0.0025 at the right are clipped in both captured frames. The visible cells are transcribed below in captured row order; unspecified row labels are not inferred.
Gallery 4.5.2, frame 1 — upper-tail probability p, captured rows 1–18
Captured rowp=.2p=.15p=.1p=.05p=.025p=.02p=.01p=.005p=.0025
11.3761.9633.0786.31412.7115.8931.8263.66127.3
21.0611.3861.8862.924.3034.8496.9659.92514.09
30.9781.251.6382.3533.1823.4824.5415.8417.453
40.9411.191.5332.1322.7762.9993.7474.6045.598
50.921.1561.4762.0152.5712.7573.3654.0324.773
60.9061.1341.441.9432.4472.6123.1433.7074.317
70.8961.1191.4151.8952.3652.5172.9983.4994.029
80.8891.1081.3971.862.3062.4492.8963.3553.833
90.8831.11.3831.8332.2622.3982.8213.253.69
100.8791.0931.3721.8122.2282.3592.7643.1693.581
110.8761.0881.3631.7962.2012.3282.7183.1063.497
120.8731.0831.3561.7822.1792.3032.6813.0553.428
130.871.0791.351.7712.162.2822.653.0123.372
140.8681.0761.3451.7612.1452.2642.6242.9773.326
150.8661.0741.3411.7532.1312.2492.6022.9473.286
160.8651.0711.3371.7462.122.2352.5832.9213.252
170.8631.0691.3331.742.112.2242.5672.8983.222
180.8621.0671.331.7342.1012.2142.5522.8783.197
Gallery 4.5.2, frame 2 — upper-tail probability p, captured rows 1–19
Captured rowp=.2p=.15p=.1p=.05p=.025p=.02p=.01p=.005p=.0025
10.8611.0661.3281.7292.0932.2052.5392.8613.174
20.861.0641.3251.7252.0862.1972.5282.8453.153
30.8591.0631.3231.7212.082.1892.5182.8313.135
40.8581.0611.3211.7172.0742.1832.5082.8193.119
50.8581.061.3191.7142.0692.1772.52.8073.104
60.8571.0591.3181.7112.0642.1722.4922.7973.091
70.8561.0581.3161.7082.062.1672.4852.7873.078
80.8561.0581.3151.7062.0562.1622.4792.7793.067
90.8551.0571.3141.7032.0522.1582.4732.7713.057
100.8551.0561.3131.7012.0482.1542.4672.7633.047
110.8541.0551.3111.6992.0452.152.4622.7563.038
120.8541.0551.311.6972.0422.1472.4572.753.03
130.8511.051.3031.6842.0212.1232.4232.7042.971
140.8491.0471.2991.6762.0092.1092.4032.6782.937
150.8481.0451.2961.67122.0992.392.662.915
160.8461.0431.2921.6641.992.0882.3742.6392.887
170.8451.0421.291.661.9842.0812.3642.6262.871
180.8421.0371.2821.6461.9622.0562.332.5812.813
190.8411.0361.2821.6451.962.0542.3262.5762.807

Confidence Level C, in the visible column order: 60%, 70%, 80%, 90%, 95%, 96%, 98%, 99%, 99.50%.

Example 4.5.1

Recall in Chapter 1.1 that a local candy company wanted to find out how many boxes of each of their nine flavors had been sold in a week. The company now wants to create a perfume line out of its flavors. One flavor in particular, “Omnom Orange”, is said to last for 12 hours. To test this, the company takes a random sample of 16 people out of thousands from their focus group. The results are shown in the following table:

Longevity of Perfume (Hours)
11.1413.0612.1113.25
13.3312.3611.7713.69
12.5611.8313.4212.29
12.9912.0514.0111.25

(a) Is there sufficient evidence at the α = 0.05 level to conclude that the perfume doesn’t last 12 hours?

(b) Construct a confidence interval consistent with the results in (a).

Example 4.5.1 — solution (a)

(a) First, we want to check our three conditions:

Random: The 16 people were taken from a random sample.

Normal: According to our Normal conditions for small sample sizes, for a sample size of 16, t-procedures can be used as long is there isn’t any strong skewness or outliers. To examine these conditions, we can draw graphs that enable us to easily detect skewness and outliers. We have a few choices here, but a histogram, boxplot, and/or Normal probability plot are some of the ones most commonly.

Source gap (S0414): Source sentence truncated: the page ends “are some of the ones most commonly”; the following prose page begins “These can be seen in Gallery 4.5.3.” No completion is captured.

These can be seen in Gallery 4.5.3. The histogram is roughly symmetric with no strong skewness, the boxplot also appears to be only slightly right-skewed with no visible outliers, and the Normal probability plot is roughly linear. All three of our techniques confirm that the data has no strong skewness or outliers and thus passes the condition.

Independent: We are sampling without replacement, so the focus group has to have at least 10(16)=160 people in it, and indeed it does since it was mentioned in the problem that it contained thousands of participants.

Since we want to know if the mean number of hours is different than 12 (not specifically greater or lesser than), we use a two-sided test.

H0:μ=μ0,Ha:μ≠μ0H_0:\mu=\mu_0,\qquad H_a:\mu\ne\mu_0

Using the formula for the one-sample t-test, we get:

t=xˉ−μ0sx2/n=12.569−12(0.860)2/16=2.649t=\frac{\bar x-\mu_0}{\sqrt{s_x^2/n}}=\frac{12.569-12}{\sqrt{(0.860)^2/16}}=2.649

Refer to Gallery 4.5.4 for how to perform these steps on your calculator for both the “Data” and “Stats” options.

We want to compare the t-statistic to the critical t-value.

To find the critical chi-square value, we need the upper tail probability and the degrees of freedom. We have a two-sided test at α = 0.05, which means the upper tail probability is 0.025 with a df = 16 − 1 = 15. With these two pieces of information, we get t* = 2.131 from the t-table.

Source gap (S0423): Source terminology: this t-test solution calls the critical value a “critical chi-square value.” The original wording is retained.

We see that the t-statistic is greater than the critical t-value (2.649 > 2.131). Alternatively, we see that the p-value that corresponds with the t-statistic is smaller than the significance level α (0.018 < 0.05). Both results tell us that we can reject the null hypothesis and conclude that the mean number of hours the perfume lasts is different than 12.

Interactive 4.5.2 Example 4.5.1

A t density curve with a green right tail beyond t=2.649. The shaded tail is annotated p-value of interest; the t=2.649 label is annotated t-statistic of Interest. Only this annotated state is captured. Open full-size image S0427
Example 4.5.1 — solution (b)

(b) A confidence interval consistent with our results in (a) would have a confidence level of 95%. We already found this value to be t* = 2.131 in part (a) for 16 − 1 = 15 degrees of freedom.

Using the formula for the one-sample t-interval, we get:

xˉ±t∗sx2n=12.569±2.131(0.860)216=(12.11113.027)\bar x\pm t^*\sqrt{\frac{s_x^2}{n}}=12.569\pm2.131\sqrt{\frac{(0.860)^2}{16}}=(12.111\quad13.027)

Refer to Gallery 4.5.5 for how to perform these steps on your calculator.

We are 95% confident that the mean number of hours the Omnom Orange-scented perfume lasts between 12.111 and 13.027 hours. Since 12 is not included in this interval (meaning that one of the possible values the mean could take isn’t 12), this agrees with the results from the significance test in (a) where we had evidence that the perfume didn’t last 12 hours.

Pop Quiz 4.5.1

Question 1 of 5

The professor wants the class to be able to score above 70 on the math test. 6 students were chosen at random and got scores of 62, 92, 75, 68, 83, and 95. For this scenario, the hypotheses to be tested are:

Question 2 of 5

From question 1, it can be concluded that

Question 3 of 5

Using the confidence interval when conducting a two-tail hypothesis test for the population mean, we do not reject the null hypothesis if the calculated value for the population mean:

Question 4 of 5

In the t-test, the sample size is approximated by calculating the:

Question 5 of 5

Considering the following output, what is the confidence interval for the population mean?

NMeanSE MeanSt DevMinimumQ1MedianQ3Maximum
411.30830.570.06560.4201410.571.11.31.472.94
Source gap (S0436): Source output: the Minimum cell reads 410.57. The Mean cell shows 1.3083 with superscript 0.57; its meaning is not stated. No confidence level is supplied in the captured prompt.
Source gap (S0432): The five captured quiz questions show unselected choices. “Check Answer” feedback and the answer key are not captured.
Glossary and deeper reading · 16 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

  • T-table

    A t-table lists critical values for t-distributions at specified degrees of freedom and tail probabilities. Its column convention must be checked before using it for a one-sided test or a two-sided interval. For example, a two-sided 95% interval leaves 2.5% in each tail when selecting its critical value.

  • Sample statistic

    A statistic is a numerical summary calculated from sample data. Its value can change when a different sample is selected from the same population. For example, the proportion of walkers among 60 sampled students is a statistic.

    Read more: Sample statistic in the captured Statistics reading →

  • Test statistic

    A test statistic summarizes how the sample departs from the null model. Its reference distribution under the null supplies the probabilities used in testing. For example, a z-statistic expresses a sample proportion’s difference from a hypothesized proportion in null-model standard-error units.

    Read more: Test statistic in the captured Statistics reading →

  • Standard deviation

    Standard deviation measures spread around the mean using squared deviations and a square root. A sample standard deviation uses the sum of squared deviations divided by one less than the sample size before taking the square root. For example, a larger standard deviation of travel times indicates greater variability, in the original time units.

    Read more: Standard deviation in the captured Statistics reading →

  • Population

    A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.

    Read more: Population in the captured Statistics reading →

  • Mean

    The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.

    Read more: Mean in the captured Statistics reading →

  • Null hypothesis

    A null hypothesis specifies the reference claim or model used to calculate a test’s probabilities. It commonly states no population difference or a particular parameter value. For example, a null hypothesis can state that the population mean travel time equals 15 minutes.

    Read more: Null hypothesis in the captured Statistics reading →

  • Sample

    A sample is the subset of a population actually studied. How it is selected matters for the conclusions that can be drawn about the population. For example, 60 selected students may be a sample from a school’s full enrollment.

    Read more: Sample in the captured Statistics reading →

  • Alternative hypothesis

    An alternative hypothesis specifies the population difference or direction for which a test seeks evidence. It should be chosen from the research question before inspecting results. For example, an alternative can state that the population mean travel time is greater than 15 minutes.

    Read more: Alternative hypothesis in the captured Statistics reading →

  • Confidence interval

    A confidence interval is a data-based range produced by a method with a specified long-run coverage rate for a population parameter. For a fixed population, the calculated interval either contains the parameter or it does not; the confidence level describes the method over repeated samples. For example, a 95% method captures the true parameter in about 95% of repeated intervals when its assumptions hold.

    Read more: Confidence interval in the captured Statistics reading →

  • Simple random sample

    A simple random sample of a fixed size gives every possible subset of that size the same chance of selection. Giving individuals equal selection chances alone does not establish that the design is an SRS. For example, randomly choosing three identifiers without replacement from ten gives an SRS when every three-identifier set is equally likely.

    Read more: Simple random sample in the captured Statistics reading →

  • Critical value

    A critical value is a cutoff from a reference distribution chosen for a confidence level or testing rule. The relevant distribution, degrees of freedom, and one-sided or two-sided procedure determine the cutoff. For example, a symmetric 95% Normal interval uses a cutoff near 1.96 standard-error units.

    Read more: Critical value in the captured Statistics reading →

  • Degrees of freedom

    Degrees of freedom describe the independent information remaining after constraints or estimated quantities are accounted for. The value depends on the procedure, rather than always being the sample size minus 1. For example, a one-sample t procedure uses n − 1 degrees of freedom, while simple regression inference uses n − 2.

  • Central limit theorem

    Editorial qualification: The captured n ≥ 30 statement is not a universal guarantee. Under the CLT’s assumptions, the accuracy of a Normal approximation depends on the population and sample size.

    For independent identically distributed observations with finite variance, the central limit theorem explains why the standardized sample mean approaches a Normal distribution as sample size increases. There is no universal sample size that makes every population sufficiently close to Normal; skewness and extreme tails affect the approximation. For example, averaging many independent waiting times can produce a less skewed distribution of averages than the original waiting-time distribution.

    Read more: Central limit theorem in the captured Statistics reading →

  • Skewness

    Skewness describes asymmetry in a quantitative distribution’s tails. A longer right tail indicates right skew, while a longer left tail indicates left skew. For example, many short waits with a few very long waits produce right skew.

    Read more: Skewness in the captured Statistics reading →

  • Confidence level

    A confidence level is the long-run fraction of intervals from a procedure that capture the true parameter under its assumptions. Holding the data and procedure fixed, increasing the confidence level generally widens the interval. For example, a 99% interval is usually wider than a 90% interval calculated from the same sample.

    Read more: Confidence level in the captured Statistics reading →

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