Hypothesis test for mean

2 A traffic officer notes the speeds of vehicles as they pass a certain point. In the past the mean of these speeds has been \(62.3 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the standard deviation has been \(10.4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). A speed limit is introduced, and following this, the mean of the speeds of 75 randomly chosen vehicles passing the point is found to be \(59.9 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).

1. Making an assumption that should be stated, test at the \(2 \%\) significance level whether the mean speed has decreased since the introduction of the speed limit.
2. Explain whether it was necessary to use the Central Limit theorem in part (i).

7 A motorist records the time taken, \(T\) minutes, to drive a particular stretch of road on each of 64 occasions. Her results are summarised by $$\Sigma t = 876.8 , \quad \Sigma t ^ { 2 } = 12657.28$$

1. Test, at the \(5 \%\) significance level, whether the mean time for the motorist to drive the stretch of road is greater than 13.1 minutes.
2. Explain whether it is necessary to use the Central Limit Theorem in your test.

4 The continuous random variable \(X\) has mean \(\mu\) and standard deviation 45. A significance test is to be carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 230\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 230\), at the \(1 \%\) significance level. A random sample of size 50 is obtained, and the sample mean is found to be 213.4.

1. Carry out the test.
2. Explain whether it is necessary to use the Central Limit Theorem in your test.

3 Research suggests that the mean reading age of a child about to start secondary school is 10.75 . The reading ages, \(X\) years, of a random sample of 80 children who were about to start secondary school in a particular district were measured, and the results are summarised as follows. $$\mathrm { n } = 80 \quad \sum \mathrm { x } = 893 \quad \sum \mathrm { x } ^ { 2 } = 10267$$

1. Test at the \(5 \%\) significance level whether the mean reading age of children about to start secondary school in this district is not 10.75 .
2. A student wrote: "Although we do not know that the distribution of \(X\) is normal, the central limit theorem allows us to assume that it is, as the sample size is large." This statement is incorrect. Give a corrected version of the student's statement.

6. Fruit-n-Veg4U Market Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is \(0.5 \mathrm {~kg} ^ { 2 }\) and the variance of the yield for the new variety of plant is \(0.75 \mathrm {~kg} ^ { 2 }\). A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .

1. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
2. Explain the relevance of the Central Limit Theorem to the test in part (a).

1 Over time LED light bulbs gradually lose brightness. For a particular type of LED bulb, it is known that the mean reduction in brightness after 10000 hours is \(2.6 \%\). A manufacturer produces a new version of this bulb, which costs less to make, but is claimed to have the same reduction in brightness after 10000 hours as the previous version. In order to check this claim, a random sample of 10 bulbs is selected. For each bulb, the original brightness and the brightness after 10000 hours are measured, in suitable units. A spreadsheet is used to produce a \(95 \%\) confidence interval for the mean percentage reduction in brightness. A screenshot of the spreadsheet is shown in Fig. 1. \begin{table}[h] \end{table}

1. State the confidence interval in the form \(a < \mu < b\).
2. Explain whether the confidence interval suggests that the mean percentage reduction in brightness after 10000 hours is different from 2.6\%.
3. Explain how the value in cell B8 was calculated.
4. State an assumption necessary for this confidence interval to be calculated.
5. Explain the advantage of using the same bulbs for both measurements.