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AP Statistics
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If our data comes from a simple random sample (SRS) and the sample size is sufficiently large, then we know that the sampling distribution of the sample means is approximately normal with mean μ and standard deviation .
The spread of the sampling distribution depends on n and σ. σ is generally unknown and must be estimated.
NOW…THEORY ASIDE AND ONTO PRACTICE !
AP Statistics, Section 11.1 2
n
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SRS – size nNormal distribution of a populationμ and σ are unknownTo estimate σ – use “S” in its place
Then the standard error of the sample mean is
AP Statistics, Section 11.1 3
sn
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The z statistic has N (0,1)When s is substituted the distribution is no longer
normal
AP Statistics, Section 11.1 4
xz
n
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The t statistic is used when we don’t know the standard deviation of the population, and instead we use the standard deviation of the sample distribution as an estimation.
The t statistic has n-1 degrees of freedom (df).
/xts n
AP Statistics, Section 11.1 5
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Interpret the t statistic in the same way as the z statistic
There is a different distribution for every sample size.
The t statistic has n-1 degrees of freedom.
Write t (k) to represent the t distribution with k degrees of freedom.
AP Statistics, Section 11.1 6
Density curves for the t distribution are similar to the normal curve (symmetrical and bell shaped)
The spread is greater and there is more probability in the tails and less in the center.
Using s introduces more variability than sigma.
As d.f. increase, t(k) gets more normal
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In statistical tests of significance, we still have H0 and Ha.
We need to provide the mu in the calculation of the t statistic.
Looking at the t table is fundamentally different than the z table.
/xts n
AP Statistics, Section 11.1 7
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Assume SRS size n with population mean μ Confidence interval will be correct for normal
populations and approx. correct for large n.
AP Statistics, Section 11.1 8
estimate t * (SE estimate)
sCI x t * ( )n
(1 C)t* f or t(n-1)2
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Let’s suppose that Mr. Young has been told that he should mop the floor by 1:25 p.m. each day.
We collect 12 sample times with an average of 27.58 minutes after 1 p.m. and with a standard deviation of 3.848 minutes.
Find a 95% confidence interval for Mr. Young’s mopping times.
AP Statistics, Section 11.1 9
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AP Statistics, Section 11.1 10
x 27.58mins 3.848n 12df 11CL :95%
From table C:t* = 2.201
3.848CI 27.58 2.20112
CI : (25.135, 30.025)
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Step 1: Population of interest:
◦ Mr. Young’s mopping time Parameter of interest:
◦ average time of arrival to mop
Hypotheses◦ H0: µ=25 min past 1:00◦ Ha: µ>25 min past 1:00
/xts n
AP Statistics, Section 11.1 11
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We are using 1 sample t-test? Bias?
◦ SRS not stated. Proceed with caution. Independence?
◦ Population size is at least 10 times the sample size?
◦ We assume that Mr. Young has mopped on a lot of days
Normality?◦ Big sample size (> 30). No◦ Sample is somewhat normal because
the sample distribution is single peaked, no obvious outliers.
/xts n
AP Statistics, Section 11.1 12
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Calculate the test statistic, and calculate the p-value from Table C
27.58 253.848 / 122.322
( 2.322) is between .025 and .02
t
P t
AP Statistics, Section 11.1 13
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Is the t-value of 2.322 statistically significant at the 5% level? At the 1% level?
Does this test provide strong evidence that Mr. Young arrives on time to complete his mopping?
AP Statistics, Section 11.1 14
Try this exercise on your calculator using:STAT TESTS TintervalSTAT TESTS T-Test
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Wednesday: 11.6 – 11.11 Thursday: 11.13 – 11.20 Friday: T-Test Worksheet
AP Statistics, Section 11.1 15