Question

1.) A researcher claims that Americans use more consonants in their language than the vowels. To test this claim, researcher randomly selects 26 English words (please randomly generate these English words by yourself and explicitly state them in your homework as your sample). Based on this sample;

a) State your hypothesis H0 and H1 claiming that Americans use more consonants in their language.

b) State the decision rule, sample statistics, table value and test statistic in the same order as in the extra solved examples on hypothesis testing.

c) Test, at the 97% level, the null hypothesis that the process is operating correctly

Answer 1

Following are the 26 randomly selected words and the number of vowels and consonants in each word

Average number of vowels = 2.8846

Average number of consonants = 4.0385

a)
The null and alternative hypotheses are
Ho :  μ1 = μ2
Ha :  μ1 > μ2
where μ1, μ2 are the population mean number of consonants and vowels in a word respectively

b)

The sample statistics are given below :

	n	Mean (M)	Variance (SS)	Standard Deviation (SD)
Number of consonants	n1 = 26	M1 = 4.0385	SS1 = 1.6385	s1 = 1.28
Number of vowels	n2 = 26	M2 = 2.8846	SS2 = 1.5462	s2 = 1.2434

We use independent Samples T test since population standard deviation is unknown, and sample size is small

Degrees of Freedom
df1 = n1 - 1 , df2 = n2 - 1 , df = n1 + n2 - 2
df = 50

Critical value (Table Value) t-crit is found using Excel function T.INV

97 % Level of significance

implies α = 0.03

Hence

t-critical = -T.INV(0.03, 50)         (Negative sign is used since this is a right tailed test)

t-critical = 1.9244

Decision Rule:

Since this is a right tailed test, the decision rule is

Reject Ho if test statistic t > 1.9244

Using the given formulae, we get
Pooled Variance

Sp² = 1.5923

Mean Squared Error S_m1-m2

S_m1-m2 = 0.35

t-statistic

t-statistic = t = 3.2969

c)

Now,

3.2969 > 1.9244

that is, calculated test statistic > table value

Hence, as per the decision rule, we reject Ho

Conclusion
There exists enough statistical evidence at 97% level of significance to show that Americans use more consonants in their language.

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