Hypothesis Testing (3)

Question 1

One of the history teachers at a school wants to survey a sample of year 7 pupils in the school. She uses all of the pupils in her Year 7 class as her sample

Identify the population for the survey

A suitable sampling frame that she could use

Answer 1

All the year 7 pupils in the school

A list of all year 7 pupils, e.g. the school registers

Question 2

One of the history teachers at a school wants to survey a sample of year 7 pupils in the school. She uses all of the pupils in her Year 7 class as her sample

The survey has questions on a number of different topics. For each of the topics given below, state, with an explanation, whether or not her sample is likely to be biased

Pupils’ opinion on history lessons at the school
How far away from the school pupils live

Answer 2

Biased - as all of the students in the sample have the same history teacher, their opinions are likely to be similar, while students with a different teacher would have different opinions on history lessons

Not biased - the distance between a pupil’s home and the school is unlikely to be affected by which history they are in, so only pupils from one class should not introduce bias

Question 3

A film club has 250 members. 98 of them are retired, 34 are unemployed, 83 work full or part time and the rest are students. The secretary wants to survey a sample of 25 members. She uses stratified sampling to select her sample. How many people in each category should she ask?

Answer 3

Retired: (98 / 250) * 25 = 9.8 = Around 10 people
Unemployed: (34 / 250) * 25 = 3.4 = Around 3 people
In employment: (83 / 250) * 25 = 8.3 = Around 8 people
Students: ((250 - 98 - 34 - 83) / 250) * 25 = 3.5 = Around 4 people

Question 4

One year ago, 43% of customers rated a restaurant as excellent. Since then, a new chef has been employed, and the manager believes that the approval rating will have gone up. He decides to carry out a hypothesis test to test his belief. Define the null and alternative hypotheses the manager should use

Answer 4

Let p be the proportion of customers that rated the restaurant as excellent. Then H_0: p = 0.43 and H_1: p > 0.43

Question 5

Over a long period of time, the chef at an Italian restaurant has found that there is a probability of 0.2 that a customer ordering a dessert in a weekday evening will order tiramisu. He thinks that the proportion of customers ordering desserts on Saturday evenings who order tiramisu is greater than 0.2

State the name of the probability distribution that would be used in a hypothesis test for the value of p, the proportion of Saturday evening dessert eaters ordering tiramisu

Answer 5

Binomial

Question 6

Over a long period of time, the chef at an Italian restaurant has found that there is a probability of 0.2 that a customer ordering a dessert in a weekday evening will order tiramisu. He thinks that the proportion of customers ordering desserts on Saturday evenings who order tiramisu is greater than 0.2

A random sample of 20 customers who ordered a dessert on a Saturday evening was taken. 7 of these customers ordered tiramisu

Stating your hypothesis clearly, test the chef’s theory at the 5% level of significance

Find the minimum number of tiramisu orders needed for the result to be significant

Answer 6

H_0: p = 0.2
H_1: p > 0.2
X = Number of tiramisu orders in sample. 
Under H_0, X ~ B(20, 0.2)
a = 0.05 

Find the p value (the probability of getting a value greater than or equal to 7, under H_0:
P(X >= 7) = 1 - P(X <= 6) = 1 - 0.9133 = 0.0867
0.0867 > 0.05, so the result isn’t significant

So there is insufficient evidence at the 5% level of significance to support the chef’s theory that the proportion of dessert eaters ordering tiramisu on a Saturday is greater than on weekdays.

ii) You’re looking for the smallest value of x such that P(X >= x) <= 0.05
You know X=7 isn’t significant from part (i)
Try 8: P(X >= 8) = 0.0321 < 0.05, so the answer is 8 tiramisu orders

Question 7

The heights of trees in an area of woodland are known to be normally distributed with a mean of 5.1m and a variance of 0.2. A random sample of 100 trees from a second area of woodland is selected and the heights, X, of the trees are measured giving the following result:
Σx = 490

Calculate the sample mean, x̄, for the trees in this second area.

Answer 7

x̄ = Σx/n = 490/100 = 4.9m

Let μ = mean height of trees in 2nd area
H_0: μ = 5.1 and H_1: μ != 5.1

Under H_0: x̄ ~ N(5.1, (0.2/100))
Z = (4.9 - 5.1)/(¬(0.2/100)) = -4.4721…

This is a two-tailed test at the 1% level, so the critical values you need are z such that P(Z < z) = 0.005 and P(Z > z) = 0.005.
Looking at these in the percentage points table gives critical values of -2.5758 and 2.5758.

Since -4.4721… < -2.5758, the result is significant.

There is evidence at the 1% level of significance to reject H_0 and to suggest that the trees in the second area have a different mean height