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## Unit 7 mean and standard deviation assignment

Question 3 (Unit 7) –14marks

(a) For the normal distribution shown in Figure 1, ﬁnd approximate values for its mean and standard deviation. Explain how you obtained your answers. [3]

(b)

(i) Firstly, the normal distribution of a variable x has mean µ =12and standard deviation σ =4. Make a hand-drawn sketch of this distribution. [3]

(ii) Secondly, 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. [1]

(iii) Thirdly, 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 ]

### Unit 7 mean and standard deviation assignment

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) Firstly, name the hypothesis test that is appropriate to use in this situation. [1]

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

(c) Thirdly, calculate, by hand, the value of the estimated standard error of the diﬀerence between the sample means. [3]

(d) Fourthly, calculate, by hand, the value of the test statistic. [2]

(e) Lastly, 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. [3]