One-sample and

Two-sample t-test

When population standard deviation (∂) is NOT KNOWN

When ∂ ia NOT given, we use the

t-statistic

____t =

x- µ

s/√n

Example: performing a one-sample t-test

Diet colas use artificial sweeteners to avoid sugar. These sweeteners gradually lose their sweetness over time. Manufacturers therefore test new colas for loss of sweetness before marketing them. Trained tasters sip the cola along with drinks of standard sweetness and score the cola on a “sweetness scale” of 1 to 10. The cola is then stored for a month at high temperature to imitate the effect of four months' storage at room temperature. Each taster scores the cola again after storage. Our data are the differences (score before storage minus score after storage) in the tasters' scores. The bigger these differences, the bigger the loss of sweetness. Here are the sweetness losses for a new cola, as measured by 10 trained tasters:

I. Hypotheses:Ho: the mean sweetness loss is µ=0Ha: the mean sweetness loss is positive µ>0

II. Conditions:SRS: the tasters are considered to be TRAINED so we can rely that the 10 tasters are taken from random. This is a matter of judgementNormality: the boxplot shows skewness so we will proceed with caution.Independence: since N≥100 taken from the entire population, we can view these samples as independent.

III CalculationsOne-sample t-test

1.02 - 0

1.196/ 10 √t = = 2.70

p-value = .0123

df = 9Syntax: tcdf (t-value, n, df)

IV. ConclusionSince the p-value of .0123 is small, it gives a strong evidence against the null-hypothesis making our test significant. Therefore we have reason to believe that the mean sweetness loss is positive. However, since our sampling distribution is skewed, we will proceed with caution.

Your turn!Healthy bones Here are estimates of the daily intakes of calcium (in milligrams) for an SRS of 38 women between the ages of 18 and 24 years who participated in a study of women's bone health:

Suppose that the recommended daily allowance (RDA) of calcium for women in this age range is 1200 milligrams. Doctors involved in the study suspected that participating subjects had significantly lower calcium intakes than the RDA.Test the doctors' claim at the α = 0.05 significance level

Ho: the mean daily calcium intake is µ=1200 mg

Ha: the mean daily calcium intake is µ<1200 mgSRS: taken from SRS of 29 women ages

18-24Normality: boxplot shows distribution is skewed to the right. Independence: N≥10(39)

One-sample t-statistic: t-value = -3.95

p-value = .00017Since our p-value is less than the 5% significance level, we

will reject the null hypothesis making our test significant. Therefore the mean daily intake of calcium of women is

significantly less than the RDA recommendation. However, we will proceed with caution since our distribution is skewed.

Two-sample t-testWe hear that listening to Mozart improves students' performance on tests. Perhaps pleasant odors have a similar effect. To test this idea, 21 subjects worked a paper-and-pencil maze while wearing a mask. The mask was either unscented or carried a floral scent. The response variable is their average time on three trials. Each subject worked the maze with both masks, in a random order. Is there an improvement in time of solving the maze with the scented mask? Carry out an appropriate test to prove this claim.

Text

Ho: the mean difference between wearing scented mask and unscented mask is µ=0Ha: the mean difference between wearing

scented mask and unscented mask is positive µ>0

SRS: sample came from an SRS of 21 subjectsNormality: The boxplot shows an approx. symmetrical distributionIndependence: We are willing to assume that the

21 subjects' are independent observations. one-sample t-statistic: t-value

= .349p-value = .25

Since our p-value is greater than the 5% or even 10% significance level, we will NOT reject the null

hypothesis making our test NOT significant. Therefore the floral scented mask did not help

improve the average time in solving the maze.

homeworkRight versus left: The design of controls and instruments affects how easily people can use them. A student project investigated this effect by asking 25 right-handed students to turn a knob (with their right hands) that moved an indicator by screw action in seconds.The project designers hoped to show that right-handed people find right-hand threads easier to use. Carry out a significance test at the 5% significance level to investigate this claim.

Answer:Ho: the mean difference between the left

hand and the right hand speed is µ=0Ha: the mean difference between the left

hand and the right hand speed is µ<0

T-value = -2.9037P-value=.0039

Since our p-value is less than the 5% significance level, we will reject the null hypothesis making our

test significant. Therefore the mean difference between the left hand the the right hand is negative.

However, we will proceed with caution since our distribution is skewed.