Exercises on writing hypotheses
Below are three examples, followed by six exercises, which are intended to give you practice in writing
down hypotheses. Answers are given. If you have questions, bring them to class. The examples and exercises
are based on information found in the Statistical abstract of the United States, 120th edition, 2000, issued by
the U.S. Department of Commerce.
Examples
E.g. 1
The percentage of U.S. college students employed full-time in 1998 was 63.7% Has this
percentage increased?
There are two possibilities: “No” and “Yes.”
The culture holds that “No” means “The percentage of U.S. college students employed full-time is still
63.7%.” If it’s still 63.7%, there has been no change in it. “No change” means “no different,” which means
“is equal to.” The traditional way of writing “The percentage. . . is. . . 63.7%” is “p = 0.637”, the letter “p”
standing for “percentage” or “proportion.” The claim “p = 0.637” is your null hypothesis, as it represents
“no change” or “no difference.”
“Yes” means “The percentage of U.S. college students employed full-time has increased.” If it has
increased, it is now greater than 63.7% The traditional way of writing this is “p > 0.637”. This is your
alternative hypothesis, because if you decide the null hypothesis is wrong, what alternative do you have?
The alternative hypothesis, of course. That’s what it’s for.
The traditional symbols for the null and alternative hypotheses are, respectively, H0 and Ha . Please write
your answers this way: H0 : p = 0.637, Ha : p > 0.637. (Some people use “H1 ” for the alternative hypothesis
instead of Ha . You may do so, if you like.)
E.g. 2
In 1990, the percentage of persons in the U.S. living in rural areas was 24.8% Has this
percentage decreased?
There are two possibilities: “No” and “Yes.”
The cultural standard is that “No” means “The percentage of persons in the U.S. living in rural areas
is still 24.8%.” The traditional way of writing this is “p = 0.248”. This is your null hypothesis, because it
represents the “status quo,” or “no change,” or “no difference.”
“Yes” means “The percentage of persons in the U.S. living in rural areas has decreased.” If it has
decreased, it is now less than 24.8%. The traditional way of writing this is “p < 0.248”. This is your
alternative (or, research) hypothesis, because it bears the burden of proof.
Please write your answers this way: H0 : p = 0.248, Ha : p < 0.248.
E.g. 3
The average farm debt in the U.S. in 1997 was $86,500 per farm. Has this average
changed?
There are two possibilities: “No” and “Yes.”
“No” means “The average farm debt in the U.S. in 1997 is still $86,500 per farm.” The traditional way
of writing this is “µ = 86, 500”. To see why, read the next paragraph. Otherwise, skip the next paragraph.
When people say “average” they almost always mean “the mean,” so we’ll assume it’s the
mean farm debt. Do we use x or µ for this mean? Well, we’re talking about all the farms in
the U.S., which is evidently a population, so we’ll use µ for this mean. But do we use the µ
that stands for the mean of all sample means or the µ that stands for the mean of all individual
measurements? The claim is about mean debt per farm; since “farm” is singular, it’s about the
mean of individual measurements. Our symbol for this mean is µ.
The claim “µ = 86, 500” is your null hypothesis, because it represents the “no change,” or “no difference,”
or “the status quo.”
“Yes” means “The average farm debt in the U.S. in 1997 is no longer $86,500 per farm.” If it is no longer
$86, 500 per farm, it can be higher than $86, 500 or lower than $86, 500. The traditional way of writing this
is “µ 6= 86, 500”, because “6=” is true if µ is higher than $86, 500 or lower than this figure, either way. The
claim “µ 6= 86, 500” is your alternative hypothesis, because it represents a change from the status quo.
Please write your answers this way: H0 : µ = 86, 500, Ha : µ 6= 86, 500.
Exercises
1. In 1999, 55.1% of high school students in the U.S. played on a high school sports team. Has this
percentage changed?
2. Airline passengers understandably do not like it when their flights are canceled or do not leave or arrive
on time. In 1999, the average number of complaints about such things was 0.27 per scheduled flight.
Some people believe that airline customers complain more these days about canceled flights and late
flights. Is this true?
3. In 1998, the average attendance at orchestral concerts was 1170 people. Has this figure changed?
4. In 1990, the percentage of pilot’s licenses in the U.S. held by women was 5.8% and was on the decline.
Is this percentage lower today?
5. Dentists often have patients whose dental care is subsidized by Medicaid, a social insurance program.
On average, each dentist in the U.S. received $6,630 from Medicaid in 1996. Because there is greater
use of social insurance today, we might expect dentists to be receiving more from Medicaid now than
in 1996. Are they?
6. People register to vote, but they don’t always vote. A 1994 study asked registered voters in the U.S.
whether they had voted in the most recent election. Of those aged 21–24 years, 42% said they had
voted. Do fewer such registered voters actually vote, nowadays?
Answers
1. H0 : p = 0.551
Ha : p 6= 0.551
2. H0 : µ = 0.27
Ha : µ > 0.27
3. H0 : µ = 1170
Ha : µ 6= 1170
4. H0 : p = 0.058
Ha : p < 0.058
5. H0 : µ = 6, 630
Ha : µ > 6, 630
6. H0 : p = 0.42
Ha : p < 0.42
Please bear in mind that H0 stands for “null hypothesis” and Ha stands for “alternative hypothesis.”
The colon between the H0 or Ha and the actual equation or nonequality that expresses the hypothesis is
traditional punctuation. IT IS NOT AN “EQUALS” SIGN, but a colon.
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