# Introducing Statisitics

Question 3 (Unit 7) –14marks

• For the normal distribution shown in Figure 1, ﬁnd approximate values for its mean and standard deviation. Explain how you obtained your answers. Figure1

(b)

(i) The normal distribution of a variable x has mean µ =12and standard deviation σ =4. Make a hand-drawn sketch of this distribution. 

(ii) Write down the formula for z that converts each value of the variable x in part (b)(i) so that z follows a standard normal distribution. Your answer should contain both the general formula and the speciﬁc formula for this example. 

(iii) Calculate the value of z corresponding to x =4. Interpret this value of z in terms of the number of standard deviations x is above or below its mean.[ 3 ]

(c) Suppose that the population distribution of the lengths of female spiders of the species Sosippusﬂoridanus (native to Florida, USA)has mean 15mm and standard deviation 2.7mm.

(i) Find the standard deviation of the sampling distribution of the mean length for samplesof60femalespiders of this species.[ 2 ]

(ii) Hence give the approximate distribution of the sample mean length for samplesof60femalespiders of this species.[ 2 ]

Question 4 (Unit 7) –11marks

Children in elementary schools in a US city were given two versions of the same test, but with the order of questions arranged from easier to more diﬃcult inversion A and in reverse order in version B. A randomly selected group of 44 students were given version A; their mean grade was 83 with standard deviation 5.6. Another randomly selected group of 38 students were given version B; their mean grade was 81 with standard deviation 5.3.

A hypothesis test is to be performed to investigate whether the population mean grade of students answering version A is the same as the population mean grade of students answering version B.

(a) Name the hypothesis test that is appropriate to use in this situation. 

(b) Using appropriate notation, which you should deﬁne, specify the null and alternative hypotheses associated with the test.[ 2 ]

(c) Calculate, by hand, the value of the estimated standard error of the diﬀerence between the sample means. 

(d) Calculate, by hand, the value of the test statistic. 

(e) The sample sizes are both greaterthan25, so you can assume that when the null hypothesis is true, the test statistic follows(approximately)the standard normal distribution. Complete the hypothesis test, carefully detailing the conclusions of the test. 

Question5(Unit 8) –13marks

The manager of a famous water park attraction states that:

• 70% of the visitors on a speciﬁc day ride the water slide

• of those visitors who ride the water slide, 35% also ride the bumper boats on the same day.

(a) Let W denote the event that a visitor on that day rides the water slide, and let B denote the event that a visitor on that day rides the bumper boats. Write the information given in the two bullet points in symbolic form.[ 2 ]

(b) Calculate the probability that a randomly chosen visitor on that day rides both the water slide and the bumper boats.[ 3 ]

(c) Additional information is now given that 50% of the visitors on that day ride the bumper boats. Calculate the probability that a randomly chosen visitor who rides the bumper boats also rides the water slide on the same day. 

(d) What percentage of visitors on that day ride either the water slide or the bumper boats, or both? 

(e) For the park visitors on that day, are the events of riding the water slide and riding the bumper boats independent? Give a reason for your answer.[ 2 ]

Question 7 (Unit 9) –10marks

Figure 2 shows a scatterplot of x and y values for each of 100 data points in a sample.

Figure2 The correlation coeﬃcient associated with Figure 2 is claimed to be − 0. 115. In answering parts (a) and (b) of this question, give reasons for your answers.

(a) Is the value claimed for the correlation coeﬃcient plausible in terms of its sign? 

(b) Is the value claimed for the correlation coeﬃcient plausible in terms of its closeness to 0?[ 2 ]

In parts(c), (d)and (e) of this question, further statements are made about the relationship between x and y in Figure2. In each case, state whether or not the statement is correct, again giving a reason for each answer.

(c) If each value of the y variable is multiplied by three, then the value of the correlation coeﬃcient will also be multiplied by three. 

(d) If the values of x and y are swapped over, then the value of the correlation coeﬃcient will be the reciprocal of the value associated with Figure2.[ 2 ]

(e) Figure 2shows that the variable x causes the variable y to take the values that it does. 

Question 8 (Unit 9) –10marks

A botanist is interested in determining the average diameter of the ﬂowers of a particular plant. She decided to take a random sample of size 80 of these ﬂowers and found that the sample mean of the ﬂower diameters was 9.6cm, and the sample standard deviation was2.2cm.

(a) Calculate the value of the estimated standard error of the sample mean.[ 2 ]

(b) Calculate a 95% conﬁdence interval for the population mean of the ﬂower diameter of this particular plant. 

(c) Interpret the conﬁdence interval from part(b) in terms of all possible random samples of ﬂowers of this plant. 

(d) On the basis of the conﬁdence interval from part(b), what would have been the outcome of a z-test of the null hypothesis that the population mean of the ﬂower diameters is 11cm? Interpret the result of the test.