ford escort zx2 2000, problem starting

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when the key is turned to start, there is no turnover of the engine. the lights are on, and it has a brand new battery. however, if i hold the key in the start position for a few seconds, the engine will turn over. im wondering if it is a troubleshooting problem with the starter. my friend has an '01 Impala and it does the same thing. he took it to the dealership and got a new starter, battery, and spark plugs(with wires)and it still does it. the dealership said they didnt know what the problem was, or how to fix it. does anyone have any ideas of how to fix this problem?